Find the Function Rule table[[x,y],[1,6],[2,10.5],[3,15],[4,19.5],[5,24]]

$\begin{array}{c} x & y \\ 1 & 6 \\ 2 & 10.5 \\ 3 & 15 \\ 4 & 19.5 \\ 5 & 24 \end{array}$

Step 1

Check if the function rule is linear.

To find if the table follows a function rule, check to see if the values follow the linear form $y = a x + b$ .

$y = a x + b$

Build a set of equations from the table such that $y = a x + b$ .

$6 = a (1) + b 10.5 = a (2) + b 15 = a (3) + b 19.5 = a (4) + b 24 = a (5) + b$

Calculate the values of $a$ and $b$ .

Solve for $a$ in $6 = a + b$ .

Rewrite the equation as $a + b = 6$ .

$a + b = 6$

$10.5 = a (2) + b$

$15 = a (3) + b$

$19.5 = a (4) + b$

$24 = a (5) + b$

Subtract $b$ from both sides of the equation.

$a = 6 - b$

$10.5 = a (2) + b$

$15 = a (3) + b$

$19.5 = a (4) + b$

$24 = a (5) + b$

$a = 6 - b$

$10.5 = a (2) + b$

$15 = a (3) + b$

$19.5 = a (4) + b$

$24 = a (5) + b$

Replace all occurrences of $a$ with $6 - b$ in each equation.

Replace all occurrences of $a$ in $10.5 = a (2) + b$ with $6 - b$ .

$10.5 = (6 - b) (2) + b$

$a = 6 - b$

$15 = a (3) + b$

$19.5 = a (4) + b$

$24 = a (5) + b$

Simplify the right side.

Simplify $(6 - b) (2) + b$ .

Simplify each term.

Apply the distributive property.

$10.5 = 6 \cdot 2 - b \cdot 2 + b$

$a = 6 - b$

$15 = a (3) + b$

$19.5 = a (4) + b$

$24 = a (5) + b$

Multiply $6$ by $2$ .

$10.5 = 12 - b \cdot 2 + b$

$a = 6 - b$

$15 = a (3) + b$

$19.5 = a (4) + b$

$24 = a (5) + b$

Multiply $2$ by $- 1$ .

$10.5 = 12 - 2 b + b$

$a = 6 - b$

$15 = a (3) + b$

$19.5 = a (4) + b$

$24 = a (5) + b$

$10.5 = 12 - 2 b + b$

$a = 6 - b$

$15 = a (3) + b$

$19.5 = a (4) + b$

$24 = a (5) + b$

Add $- 2 b$ and $b$ .

$10.5 = 12 - b$

$a = 6 - b$

$15 = a (3) + b$

$19.5 = a (4) + b$

$24 = a (5) + b$

$10.5 = 12 - b$

$a = 6 - b$

$15 = a (3) + b$

$19.5 = a (4) + b$

$24 = a (5) + b$

$10.5 = 12 - b$

$a = 6 - b$

$15 = a (3) + b$

$19.5 = a (4) + b$

$24 = a (5) + b$

Replace all occurrences of $a$ in $15 = a (3) + b$ with $6 - b$ .

$15 = (6 - b) (3) + b$

$10.5 = 12 - b$

$a = 6 - b$

$19.5 = a (4) + b$

$24 = a (5) + b$

Simplify the right side.

Simplify $(6 - b) (3) + b$ .

Simplify each term.

Apply the distributive property.

$15 = 6 \cdot 3 - b \cdot 3 + b$

$10.5 = 12 - b$

$a = 6 - b$

$19.5 = a (4) + b$

$24 = a (5) + b$

Multiply $6$ by $3$ .

$15 = 18 - b \cdot 3 + b$

$10.5 = 12 - b$

$a = 6 - b$

$19.5 = a (4) + b$

$24 = a (5) + b$

Multiply $3$ by $- 1$ .

$15 = 18 - 3 b + b$

$10.5 = 12 - b$

$a = 6 - b$

$19.5 = a (4) + b$

$24 = a (5) + b$

$15 = 18 - 3 b + b$

$10.5 = 12 - b$

$a = 6 - b$

$19.5 = a (4) + b$

$24 = a (5) + b$

Add $- 3 b$ and $b$ .

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

$19.5 = a (4) + b$

$24 = a (5) + b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

$19.5 = a (4) + b$

$24 = a (5) + b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

$19.5 = a (4) + b$

$24 = a (5) + b$

Replace all occurrences of $a$ in $19.5 = a (4) + b$ with $6 - b$ .

$19.5 = (6 - b) (4) + b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

$24 = a (5) + b$

Simplify the right side.

Simplify $(6 - b) (4) + b$ .

Simplify each term.

Apply the distributive property.

$19.5 = 6 \cdot 4 - b \cdot 4 + b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

$24 = a (5) + b$

Multiply $6$ by $4$ .

$19.5 = 24 - b \cdot 4 + b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

$24 = a (5) + b$

Multiply $4$ by $- 1$ .

$19.5 = 24 - 4 b + b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

$24 = a (5) + b$

$19.5 = 24 - 4 b + b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

$24 = a (5) + b$

Add $- 4 b$ and $b$ .

$19.5 = 24 - 3 b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

$24 = a (5) + b$

$19.5 = 24 - 3 b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

$24 = a (5) + b$

$19.5 = 24 - 3 b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

$24 = a (5) + b$

Replace all occurrences of $a$ in $24 = a (5) + b$ with $6 - b$ .

$24 = (6 - b) (5) + b$

$19.5 = 24 - 3 b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

Simplify the right side.

Simplify $(6 - b) (5) + b$ .

Simplify each term.

Apply the distributive property.

$24 = 6 \cdot 5 - b \cdot 5 + b$

$19.5 = 24 - 3 b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

Multiply $6$ by $5$ .

$24 = 30 - b \cdot 5 + b$

$19.5 = 24 - 3 b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

Multiply $5$ by $- 1$ .

$24 = 30 - 5 b + b$

$19.5 = 24 - 3 b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

$24 = 30 - 5 b + b$

$19.5 = 24 - 3 b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

Add $- 5 b$ and $b$ .

$24 = 30 - 4 b$

$19.5 = 24 - 3 b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

$24 = 30 - 4 b$

$19.5 = 24 - 3 b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

$24 = 30 - 4 b$

$19.5 = 24 - 3 b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

$24 = 30 - 4 b$

$19.5 = 24 - 3 b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

Solve for $b$ in $24 = 30 - 4 b$ .

Rewrite the equation as $30 - 4 b = 24$ .

$30 - 4 b = 24$

$19.5 = 24 - 3 b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

Move all terms not containing $b$ to the right side of the equation.

Subtract $30$ from both sides of the equation.

$- 4 b = 24 - 30$

$19.5 = 24 - 3 b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

Subtract $30$ from $24$ .

$- 4 b = - 6$

$19.5 = 24 - 3 b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

$- 4 b = - 6$

$19.5 = 24 - 3 b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

Divide each term in $- 4 b = - 6$ by $- 4$ and simplify.

Divide each term in $- 4 b = - 6$ by $- 4$ .

$\frac{- 4 b}{- 4} = \frac{- 6}{- 4}$

$19.5 = 24 - 3 b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 b}{- 4} = \frac{- 6}{- 4}$

$19.5 = 24 - 3 b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

Divide $b$ by $1$ .

$b = \frac{- 6}{- 4}$

$19.5 = 24 - 3 b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

$b = \frac{- 6}{- 4}$

$19.5 = 24 - 3 b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

$b = \frac{- 6}{- 4}$

$19.5 = 24 - 3 b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

Simplify the right side.

Cancel the common factor of $- 6$ and $- 4$ .

Factor $- 2$ out of $- 6$ .

$b = \frac{- 2 \cdot 3}{- 4}$

$19.5 = 24 - 3 b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

Cancel the common factors.

Factor $- 2$ out of $- 4$ .

$b = \frac{- 2 \cdot 3}{- 2 \cdot 2}$

$19.5 = 24 - 3 b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

Cancel the common factor.

$b = \frac{- 2 \cdot 3}{- 2 \cdot 2}$

$19.5 = 24 - 3 b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

Rewrite the expression.

$b = \frac{3}{2}$

$19.5 = 24 - 3 b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

$b = \frac{3}{2}$

$19.5 = 24 - 3 b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

$b = \frac{3}{2}$

$19.5 = 24 - 3 b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

$b = \frac{3}{2}$

$19.5 = 24 - 3 b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

$b = \frac{3}{2}$

$19.5 = 24 - 3 b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

$b = \frac{3}{2}$

$19.5 = 24 - 3 b$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

Replace all occurrences of $b$ with $\frac{3}{2}$ in each equation.

Replace all occurrences of $b$ in $19.5 = 24 - 3 b$ with $\frac{3}{2}$ .

$19.5 = 24 - 3 (\frac{3}{2})$

$b = \frac{3}{2}$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

Simplify the right side.

Simplify $24 - 3 (\frac{3}{2})$ .

Simplify each term.

Multiply $- 3 (\frac{3}{2})$ .

Combine $- 3$ and $\frac{3}{2}$ .

$19.5 = 24 + \frac{- 3 \cdot 3}{2}$

$b = \frac{3}{2}$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

Multiply $- 3$ by $3$ .

$19.5 = 24 + \frac{- 9}{2}$

$b = \frac{3}{2}$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

$19.5 = 24 + \frac{- 9}{2}$

$b = \frac{3}{2}$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

Move the negative in front of the fraction.

$19.5 = 24 - \frac{9}{2}$

$b = \frac{3}{2}$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

$19.5 = 24 - \frac{9}{2}$

$b = \frac{3}{2}$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

To write $24$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$19.5 = 24 \cdot \frac{2}{2} - \frac{9}{2}$

$b = \frac{3}{2}$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

Combine $24$ and $\frac{2}{2}$ .

$19.5 = \frac{24 \cdot 2}{2} - \frac{9}{2}$

$b = \frac{3}{2}$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

Combine the numerators over the common denominator.

$19.5 = \frac{24 \cdot 2 - 9}{2}$

$b = \frac{3}{2}$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

Simplify the numerator.

Multiply $24$ by $2$ .

$19.5 = \frac{48 - 9}{2}$

$b = \frac{3}{2}$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

Subtract $9$ from $48$ .

$19.5 = \frac{39}{2}$

$b = \frac{3}{2}$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

$19.5 = \frac{39}{2}$

$b = \frac{3}{2}$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

$19.5 = \frac{39}{2}$

$b = \frac{3}{2}$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

$19.5 = \frac{39}{2}$

$b = \frac{3}{2}$

$15 = 18 - 2 b$

$10.5 = 12 - b$

$a = 6 - b$

Replace all occurrences of $b$ in $15 = 18 - 2 b$ with $\frac{3}{2}$ .

$15 = 18 - 2 (\frac{3}{2})$

$19.5 = \frac{39}{2}$

$b = \frac{3}{2}$

$10.5 = 12 - b$

$a = 6 - b$

Simplify the right side.

Simplify $18 - 2 (\frac{3}{2})$ .

Simplify each term.

Cancel the common factor of $2$ .

Factor $2$ out of $- 2$ .

$15 = 18 + 2 (- 1) (\frac{3}{2})$

$19.5 = \frac{39}{2}$

$b = \frac{3}{2}$

$10.5 = 12 - b$

$a = 6 - b$

Cancel the common factor.

$15 = 18 + 2 \cdot (- 1 (\frac{3}{2}))$

$19.5 = \frac{39}{2}$

$b = \frac{3}{2}$

$10.5 = 12 - b$

$a = 6 - b$

Rewrite the expression.

$15 = 18 - 1 \cdot 3$

$19.5 = \frac{39}{2}$

$b = \frac{3}{2}$

$10.5 = 12 - b$

$a = 6 - b$

$15 = 18 - 1 \cdot 3$

$19.5 = \frac{39}{2}$

$b = \frac{3}{2}$

$10.5 = 12 - b$

$a = 6 - b$

Multiply $- 1$ by $3$ .

$15 = 18 - 3$

$19.5 = \frac{39}{2}$

$b = \frac{3}{2}$

$10.5 = 12 - b$

$a = 6 - b$

$15 = 18 - 3$

$19.5 = \frac{39}{2}$

$b = \frac{3}{2}$

$10.5 = 12 - b$

$a = 6 - b$

Subtract $3$ from $18$ .

$15 = 15$

$19.5 = \frac{39}{2}$

$b = \frac{3}{2}$

$10.5 = 12 - b$

$a = 6 - b$

$15 = 15$

$19.5 = \frac{39}{2}$

$b = \frac{3}{2}$

$10.5 = 12 - b$

$a = 6 - b$

$15 = 15$

$19.5 = \frac{39}{2}$

$b = \frac{3}{2}$

$10.5 = 12 - b$

$a = 6 - b$

Replace all occurrences of $b$ in $10.5 = 12 - b$ with $\frac{3}{2}$ .

$10.5 = 12 - (\frac{3}{2})$

$15 = 15$

$19.5 = \frac{39}{2}$

$b = \frac{3}{2}$

$a = 6 - b$

Simplify the right side.

Simplify $12 - (\frac{3}{2})$ .

To write $12$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$10.5 = 12 \cdot \frac{2}{2} - \frac{3}{2}$

$15 = 15$

$19.5 = \frac{39}{2}$

$b = \frac{3}{2}$

$a = 6 - b$

Combine $12$ and $\frac{2}{2}$ .

$10.5 = \frac{12 \cdot 2}{2} - \frac{3}{2}$

$15 = 15$

$19.5 = \frac{39}{2}$

$b = \frac{3}{2}$

$a = 6 - b$

Combine the numerators over the common denominator.

$10.5 = \frac{12 \cdot 2 - 3}{2}$

$15 = 15$

$19.5 = \frac{39}{2}$

$b = \frac{3}{2}$

$a = 6 - b$

Simplify the numerator.

Multiply $12$ by $2$ .

$10.5 = \frac{24 - 3}{2}$

$15 = 15$

$19.5 = \frac{39}{2}$

$b = \frac{3}{2}$

$a = 6 - b$

Subtract $3$ from $24$ .

$10.5 = \frac{21}{2}$

$15 = 15$

$19.5 = \frac{39}{2}$

$b = \frac{3}{2}$

$a = 6 - b$

$10.5 = \frac{21}{2}$

$15 = 15$

$19.5 = \frac{39}{2}$

$b = \frac{3}{2}$

$a = 6 - b$

$10.5 = \frac{21}{2}$

$15 = 15$

$19.5 = \frac{39}{2}$

$b = \frac{3}{2}$

$a = 6 - b$

$10.5 = \frac{21}{2}$

$15 = 15$

$19.5 = \frac{39}{2}$

$b = \frac{3}{2}$

$a = 6 - b$

Replace all occurrences of $b$ in $a = 6 - b$ with $\frac{3}{2}$ .

$a = 6 - (\frac{3}{2})$

$10.5 = \frac{21}{2}$

$15 = 15$

$19.5 = \frac{39}{2}$

$b = \frac{3}{2}$

Simplify the right side.

Simplify $6 - (\frac{3}{2})$ .

To write $6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$a = 6 \cdot \frac{2}{2} - \frac{3}{2}$

$10.5 = \frac{21}{2}$

$15 = 15$

$19.5 = \frac{39}{2}$

$b = \frac{3}{2}$

Combine $6$ and $\frac{2}{2}$ .

$a = \frac{6 \cdot 2}{2} - \frac{3}{2}$

$10.5 = \frac{21}{2}$

$15 = 15$

$19.5 = \frac{39}{2}$

$b = \frac{3}{2}$

Combine the numerators over the common denominator.

$a = \frac{6 \cdot 2 - 3}{2}$

$10.5 = \frac{21}{2}$

$15 = 15$

$19.5 = \frac{39}{2}$

$b = \frac{3}{2}$

Simplify the numerator.

Multiply $6$ by $2$ .

$a = \frac{12 - 3}{2}$

$10.5 = \frac{21}{2}$

$15 = 15$

$19.5 = \frac{39}{2}$

$b = \frac{3}{2}$

Subtract $3$ from $12$ .

$a = \frac{9}{2}$

$10.5 = \frac{21}{2}$

$15 = 15$

$19.5 = \frac{39}{2}$

$b = \frac{3}{2}$

$a = \frac{9}{2}$

$10.5 = \frac{21}{2}$

$15 = 15$

$19.5 = \frac{39}{2}$

$b = \frac{3}{2}$

$a = \frac{9}{2}$

$10.5 = \frac{21}{2}$

$15 = 15$

$19.5 = \frac{39}{2}$

$b = \frac{3}{2}$

$a = \frac{9}{2}$

$10.5 = \frac{21}{2}$

$15 = 15$

$19.5 = \frac{39}{2}$

$b = \frac{3}{2}$

$a = \frac{9}{2}$

$10.5 = \frac{21}{2}$

$15 = 15$

$19.5 = \frac{39}{2}$

$b = \frac{3}{2}$

Since $10.5 = \frac{21}{2}$ is not true, there is no solution.

No solution

Since $y \neq y$ for the corresponding $x$ values, the function is not linear.

The function is not linear

Step 2

Check if the function rule is quadratic.

To find if the table follows a function rule, check whether the function rule could follow the form $y = a x^{2} + b x + c$ .

$y = a x^{2} + b x + c$

Build a set of $3$ equations from the table such that $y = a x^{2} + b x + c$ .

Calculate the values of $a$ , $b$ , and $c$ .

Solve for $a$ in $6 = a + b + c$ .

Rewrite the equation as $a + b + c = 6$ .

$a + b + c = 6$

$10.5 = a \cdot 2^{2} + b (2) + c$

$15 = a \cdot 3^{2} + b (3) + c$

$19.5 = a \cdot 4^{2} + b (4) + c$

$24 = a \cdot 5^{2} + b (5) + c$

Move all terms not containing $a$ to the right side of the equation.

Subtract $b$ from both sides of the equation.

$a + c = 6 - b$

$10.5 = a \cdot 2^{2} + b (2) + c$

$15 = a \cdot 3^{2} + b (3) + c$

$19.5 = a \cdot 4^{2} + b (4) + c$

$24 = a \cdot 5^{2} + b (5) + c$

Subtract $c$ from both sides of the equation.

$a = 6 - b - c$

$10.5 = a \cdot 2^{2} + b (2) + c$

$15 = a \cdot 3^{2} + b (3) + c$

$19.5 = a \cdot 4^{2} + b (4) + c$

$24 = a \cdot 5^{2} + b (5) + c$

$a = 6 - b - c$

$10.5 = a \cdot 2^{2} + b (2) + c$

$15 = a \cdot 3^{2} + b (3) + c$

$19.5 = a \cdot 4^{2} + b (4) + c$

$24 = a \cdot 5^{2} + b (5) + c$

$a = 6 - b - c$

$10.5 = a \cdot 2^{2} + b (2) + c$

$15 = a \cdot 3^{2} + b (3) + c$

$19.5 = a \cdot 4^{2} + b (4) + c$

$24 = a \cdot 5^{2} + b (5) + c$

Replace all occurrences of $a$ with $6 - b - c$ in each equation.

Replace all occurrences of $a$ in $10.5 = a \cdot 2^{2} + b (2) + c$ with $6 - b - c$ .

$10.5 = (6 - b - c) \cdot 2^{2} + b (2) + c$

$a = 6 - b - c$

$15 = a \cdot 3^{2} + b (3) + c$

$19.5 = a \cdot 4^{2} + b (4) + c$

$24 = a \cdot 5^{2} + b (5) + c$

Simplify the right side.

Simplify $(6 - b - c) \cdot 2^{2} + b (2) + c$ .

Simplify each term.

Raise $2$ to the power of $2$ .

$10.5 = (6 - b - c) \cdot 4 + b (2) + c$

$a = 6 - b - c$

$15 = a \cdot 3^{2} + b (3) + c$

$19.5 = a \cdot 4^{2} + b (4) + c$

$24 = a \cdot 5^{2} + b (5) + c$

Apply the distributive property.

$10.5 = 6 \cdot 4 - b \cdot 4 - c \cdot 4 + b (2) + c$

$a = 6 - b - c$

$15 = a \cdot 3^{2} + b (3) + c$

$19.5 = a \cdot 4^{2} + b (4) + c$

$24 = a \cdot 5^{2} + b (5) + c$

Simplify.

Multiply $6$ by $4$ .

$10.5 = 24 - b \cdot 4 - c \cdot 4 + b (2) + c$

$a = 6 - b - c$

$15 = a \cdot 3^{2} + b (3) + c$

$19.5 = a \cdot 4^{2} + b (4) + c$

$24 = a \cdot 5^{2} + b (5) + c$

Multiply $4$ by $- 1$ .

$10.5 = 24 - 4 b - c \cdot 4 + b (2) + c$

$a = 6 - b - c$

$15 = a \cdot 3^{2} + b (3) + c$

$19.5 = a \cdot 4^{2} + b (4) + c$

$24 = a \cdot 5^{2} + b (5) + c$

Multiply $4$ by $- 1$ .

$10.5 = 24 - 4 b - 4 c + b (2) + c$

$a = 6 - b - c$

$15 = a \cdot 3^{2} + b (3) + c$

$19.5 = a \cdot 4^{2} + b (4) + c$

$24 = a \cdot 5^{2} + b (5) + c$

$10.5 = 24 - 4 b - 4 c + b (2) + c$

$a = 6 - b - c$

$15 = a \cdot 3^{2} + b (3) + c$

$19.5 = a \cdot 4^{2} + b (4) + c$

$24 = a \cdot 5^{2} + b (5) + c$

Move $2$ to the left of $b$ .

$10.5 = 24 - 4 b - 4 c + 2 b + c$

$a = 6 - b - c$

$15 = a \cdot 3^{2} + b (3) + c$

$19.5 = a \cdot 4^{2} + b (4) + c$

$24 = a \cdot 5^{2} + b (5) + c$

$10.5 = 24 - 4 b - 4 c + 2 b + c$

$a = 6 - b - c$

$15 = a \cdot 3^{2} + b (3) + c$

$19.5 = a \cdot 4^{2} + b (4) + c$

$24 = a \cdot 5^{2} + b (5) + c$

Simplify by adding terms.

Add $- 4 b$ and $2 b$ .

$10.5 = 24 - 2 b - 4 c + c$

$a = 6 - b - c$

$15 = a \cdot 3^{2} + b (3) + c$

$19.5 = a \cdot 4^{2} + b (4) + c$

$24 = a \cdot 5^{2} + b (5) + c$

Add $- 4 c$ and $c$ .

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$15 = a \cdot 3^{2} + b (3) + c$

$19.5 = a \cdot 4^{2} + b (4) + c$

$24 = a \cdot 5^{2} + b (5) + c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$15 = a \cdot 3^{2} + b (3) + c$

$19.5 = a \cdot 4^{2} + b (4) + c$

$24 = a \cdot 5^{2} + b (5) + c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$15 = a \cdot 3^{2} + b (3) + c$

$19.5 = a \cdot 4^{2} + b (4) + c$

$24 = a \cdot 5^{2} + b (5) + c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$15 = a \cdot 3^{2} + b (3) + c$

$19.5 = a \cdot 4^{2} + b (4) + c$

$24 = a \cdot 5^{2} + b (5) + c$

Replace all occurrences of $a$ in $15 = a \cdot 3^{2} + b (3) + c$ with $6 - b - c$ .

$15 = (6 - b - c) \cdot 3^{2} + b (3) + c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$19.5 = a \cdot 4^{2} + b (4) + c$

$24 = a \cdot 5^{2} + b (5) + c$

Simplify the right side.

Simplify $(6 - b - c) \cdot 3^{2} + b (3) + c$ .

Simplify each term.

Raise $3$ to the power of $2$ .

$15 = (6 - b - c) \cdot 9 + b (3) + c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$19.5 = a \cdot 4^{2} + b (4) + c$

$24 = a \cdot 5^{2} + b (5) + c$

Apply the distributive property.

$15 = 6 \cdot 9 - b \cdot 9 - c \cdot 9 + b (3) + c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$19.5 = a \cdot 4^{2} + b (4) + c$

$24 = a \cdot 5^{2} + b (5) + c$

Simplify.

Multiply $6$ by $9$ .

$15 = 54 - b \cdot 9 - c \cdot 9 + b (3) + c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$19.5 = a \cdot 4^{2} + b (4) + c$

$24 = a \cdot 5^{2} + b (5) + c$

Multiply $9$ by $- 1$ .

$15 = 54 - 9 b - c \cdot 9 + b (3) + c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$19.5 = a \cdot 4^{2} + b (4) + c$

$24 = a \cdot 5^{2} + b (5) + c$

Multiply $9$ by $- 1$ .

$15 = 54 - 9 b - 9 c + b (3) + c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$19.5 = a \cdot 4^{2} + b (4) + c$

$24 = a \cdot 5^{2} + b (5) + c$

$15 = 54 - 9 b - 9 c + b (3) + c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$19.5 = a \cdot 4^{2} + b (4) + c$

$24 = a \cdot 5^{2} + b (5) + c$

Move $3$ to the left of $b$ .

$15 = 54 - 9 b - 9 c + 3 b + c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$19.5 = a \cdot 4^{2} + b (4) + c$

$24 = a \cdot 5^{2} + b (5) + c$

$15 = 54 - 9 b - 9 c + 3 b + c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$19.5 = a \cdot 4^{2} + b (4) + c$

$24 = a \cdot 5^{2} + b (5) + c$

Simplify by adding terms.

Add $- 9 b$ and $3 b$ .

$15 = 54 - 6 b - 9 c + c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$19.5 = a \cdot 4^{2} + b (4) + c$

$24 = a \cdot 5^{2} + b (5) + c$

Add $- 9 c$ and $c$ .

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$19.5 = a \cdot 4^{2} + b (4) + c$

$24 = a \cdot 5^{2} + b (5) + c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$19.5 = a \cdot 4^{2} + b (4) + c$

$24 = a \cdot 5^{2} + b (5) + c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$19.5 = a \cdot 4^{2} + b (4) + c$

$24 = a \cdot 5^{2} + b (5) + c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$19.5 = a \cdot 4^{2} + b (4) + c$

$24 = a \cdot 5^{2} + b (5) + c$

Replace all occurrences of $a$ in $19.5 = a \cdot 4^{2} + b (4) + c$ with $6 - b - c$ .

$19.5 = (6 - b - c) \cdot 4^{2} + b (4) + c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$24 = a \cdot 5^{2} + b (5) + c$

Simplify the right side.

Simplify $(6 - b - c) \cdot 4^{2} + b (4) + c$ .

Simplify each term.

Raise $4$ to the power of $2$ .

$19.5 = (6 - b - c) \cdot 16 + b (4) + c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$24 = a \cdot 5^{2} + b (5) + c$

Apply the distributive property.

$19.5 = 6 \cdot 16 - b \cdot 16 - c \cdot 16 + b (4) + c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$24 = a \cdot 5^{2} + b (5) + c$

Simplify.

Multiply $6$ by $16$ .

$19.5 = 96 - b \cdot 16 - c \cdot 16 + b (4) + c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$24 = a \cdot 5^{2} + b (5) + c$

Multiply $16$ by $- 1$ .

$19.5 = 96 - 16 b - c \cdot 16 + b (4) + c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$24 = a \cdot 5^{2} + b (5) + c$

Multiply $16$ by $- 1$ .

$19.5 = 96 - 16 b - 16 c + b (4) + c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$24 = a \cdot 5^{2} + b (5) + c$

$19.5 = 96 - 16 b - 16 c + b (4) + c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$24 = a \cdot 5^{2} + b (5) + c$

Move $4$ to the left of $b$ .

$19.5 = 96 - 16 b - 16 c + 4 b + c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$24 = a \cdot 5^{2} + b (5) + c$

$19.5 = 96 - 16 b - 16 c + 4 b + c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$24 = a \cdot 5^{2} + b (5) + c$

Simplify by adding terms.

Add $- 16 b$ and $4 b$ .

$19.5 = 96 - 12 b - 16 c + c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$24 = a \cdot 5^{2} + b (5) + c$

Add $- 16 c$ and $c$ .

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$24 = a \cdot 5^{2} + b (5) + c$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$24 = a \cdot 5^{2} + b (5) + c$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$24 = a \cdot 5^{2} + b (5) + c$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$24 = a \cdot 5^{2} + b (5) + c$

Replace all occurrences of $a$ in $24 = a \cdot 5^{2} + b (5) + c$ with $6 - b - c$ .

$24 = (6 - b - c) \cdot 5^{2} + b (5) + c$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Simplify the right side.

Simplify $(6 - b - c) \cdot 5^{2} + b (5) + c$ .

Simplify each term.

Raise $5$ to the power of $2$ .

$24 = (6 - b - c) \cdot 25 + b (5) + c$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Apply the distributive property.

$24 = 6 \cdot 25 - b \cdot 25 - c \cdot 25 + b (5) + c$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Simplify.

Multiply $6$ by $25$ .

$24 = 150 - b \cdot 25 - c \cdot 25 + b (5) + c$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Multiply $25$ by $- 1$ .

$24 = 150 - 25 b - c \cdot 25 + b (5) + c$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Multiply $25$ by $- 1$ .

$24 = 150 - 25 b - 25 c + b (5) + c$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$24 = 150 - 25 b - 25 c + b (5) + c$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Move $5$ to the left of $b$ .

$24 = 150 - 25 b - 25 c + 5 b + c$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$24 = 150 - 25 b - 25 c + 5 b + c$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Simplify by adding terms.

Add $- 25 b$ and $5 b$ .

$24 = 150 - 20 b - 25 c + c$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Add $- 25 c$ and $c$ .

$24 = 150 - 20 b - 24 c$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$24 = 150 - 20 b - 24 c$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$24 = 150 - 20 b - 24 c$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$24 = 150 - 20 b - 24 c$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$24 = 150 - 20 b - 24 c$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Solve for $b$ in $24 = 150 - 20 b - 24 c$ .

Rewrite the equation as $150 - 20 b - 24 c = 24$ .

$150 - 20 b - 24 c = 24$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Move all terms not containing $b$ to the right side of the equation.

Subtract $150$ from both sides of the equation.

$- 20 b - 24 c = 24 - 150$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Add $24 c$ to both sides of the equation.

$- 20 b = 24 - 150 + 24 c$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Subtract $150$ from $24$ .

$- 20 b = - 126 + 24 c$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$- 20 b = - 126 + 24 c$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Divide each term in $- 20 b = - 126 + 24 c$ by $- 20$ and simplify.

Divide each term in $- 20 b = - 126 + 24 c$ by $- 20$ .

$\frac{- 20 b}{- 20} = \frac{- 126}{- 20} + \frac{24 c}{- 20}$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Simplify the left side.

Cancel the common factor of $- 20$ .

Cancel the common factor.

$\frac{- 20 b}{- 20} = \frac{- 126}{- 20} + \frac{24 c}{- 20}$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Divide $b$ by $1$ .

$b = \frac{- 126}{- 20} + \frac{24 c}{- 20}$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$b = \frac{- 126}{- 20} + \frac{24 c}{- 20}$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$b = \frac{- 126}{- 20} + \frac{24 c}{- 20}$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 126$ and $- 20$ .

Factor $- 2$ out of $- 126$ .

$b = \frac{- 2 \cdot 63}{- 20} + \frac{24 c}{- 20}$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Cancel the common factors.

Factor $- 2$ out of $- 20$ .

$b = \frac{- 2 \cdot 63}{- 2 \cdot 10} + \frac{24 c}{- 20}$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Cancel the common factor.

$b = \frac{- 2 \cdot 63}{- 2 \cdot 10} + \frac{24 c}{- 20}$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Rewrite the expression.

$b = \frac{63}{10} + \frac{24 c}{- 20}$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$b = \frac{63}{10} + \frac{24 c}{- 20}$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$b = \frac{63}{10} + \frac{24 c}{- 20}$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Cancel the common factor of $24$ and $- 20$ .

Factor $4$ out of $24 c$ .

$b = \frac{63}{10} + \frac{4 (6 c)}{- 20}$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Cancel the common factors.

Factor $4$ out of $- 20$ .

$b = \frac{63}{10} + \frac{4 (6 c)}{4 (- 5)}$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Cancel the common factor.

$b = \frac{63}{10} + \frac{4 (6 c)}{4 \cdot - 5}$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Rewrite the expression.

$b = \frac{63}{10} + \frac{6 c}{- 5}$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$b = \frac{63}{10} + \frac{6 c}{- 5}$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$b = \frac{63}{10} + \frac{6 c}{- 5}$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Move the negative in front of the fraction.

$b = \frac{63}{10} - \frac{6 c}{5}$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$b = \frac{63}{10} - \frac{6 c}{5}$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$b = \frac{63}{10} - \frac{6 c}{5}$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$b = \frac{63}{10} - \frac{6 c}{5}$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$b = \frac{63}{10} - \frac{6 c}{5}$

$19.5 = 96 - 12 b - 15 c$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Replace all occurrences of $b$ with $\frac{63}{10} - \frac{6 c}{5}$ in each equation.

Replace all occurrences of $b$ in $19.5 = 96 - 12 b - 15 c$ with $\frac{63}{10} - \frac{6 c}{5}$ .

$19.5 = 96 - 12 (\frac{63}{10} - \frac{6 c}{5}) - 15 c$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Simplify the right side.

Simplify $96 - 12 (\frac{63}{10} - \frac{6 c}{5}) - 15 c$ .

Simplify each term.

Apply the distributive property.

$19.5 = 96 - 12 (\frac{63}{10}) - 12 (- \frac{6 c}{5}) - 15 c$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Cancel the common factor of $2$ .

Factor $2$ out of $- 12$ .

$19.5 = 96 + 2 (- 6) (\frac{63}{10}) - 12 (- \frac{6 c}{5}) - 15 c$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Factor $2$ out of $10$ .

$19.5 = 96 + 2 \cdot (- 6 \frac{63}{2 \cdot 5}) - 12 (- \frac{6 c}{5}) - 15 c$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Cancel the common factor.

$19.5 = 96 + 2 \cdot (- 6 \frac{63}{2 \cdot 5}) - 12 (- \frac{6 c}{5}) - 15 c$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Rewrite the expression.

$19.5 = 96 - 6 (\frac{63}{5}) - 12 (- \frac{6 c}{5}) - 15 c$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$19.5 = 96 - 6 (\frac{63}{5}) - 12 (- \frac{6 c}{5}) - 15 c$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Combine $- 6$ and $\frac{63}{5}$ .

$19.5 = 96 + \frac{- 6 \cdot 63}{5} - 12 (- \frac{6 c}{5}) - 15 c$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Multiply $- 6$ by $63$ .

$19.5 = 96 + \frac{- 378}{5} - 12 (- \frac{6 c}{5}) - 15 c$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Multiply $- 12 (- \frac{6 c}{5})$ .

Multiply $- 1$ by $- 12$ .

$19.5 = 96 + \frac{- 378}{5} + 12 (\frac{6 c}{5}) - 15 c$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Combine $12$ and $\frac{6 c}{5}$ .

$19.5 = 96 + \frac{- 378}{5} + \frac{12 (6 c)}{5} - 15 c$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Multiply $6$ by $12$ .

$19.5 = 96 + \frac{- 378}{5} + \frac{72 c}{5} - 15 c$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$19.5 = 96 + \frac{- 378}{5} + \frac{72 c}{5} - 15 c$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Move the negative in front of the fraction.

$19.5 = 96 - \frac{378}{5} + \frac{72 c}{5} - 15 c$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$19.5 = 96 - \frac{378}{5} + \frac{72 c}{5} - 15 c$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

To write $96$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$19.5 = 96 \cdot \frac{5}{5} - \frac{378}{5} + \frac{72 c}{5} - 15 c$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Combine $96$ and $\frac{5}{5}$ .

$19.5 = \frac{96 \cdot 5}{5} - \frac{378}{5} + \frac{72 c}{5} - 15 c$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Combine the numerators over the common denominator.

$19.5 = \frac{96 \cdot 5 - 378}{5} + \frac{72 c}{5} - 15 c$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Simplify the numerator.

Multiply $96$ by $5$ .

$19.5 = \frac{480 - 378}{5} + \frac{72 c}{5} - 15 c$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Subtract $378$ from $480$ .

$19.5 = \frac{102}{5} + \frac{72 c}{5} - 15 c$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$19.5 = \frac{102}{5} + \frac{72 c}{5} - 15 c$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

To write $- 15 c$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$19.5 = \frac{102}{5} + \frac{72 c}{5} - 15 c \cdot \frac{5}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Combine $- 15 c$ and $\frac{5}{5}$ .

$19.5 = \frac{102}{5} + \frac{72 c}{5} + \frac{- 15 c \cdot 5}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Combine the numerators over the common denominator.

$19.5 = \frac{102}{5} + \frac{72 c - 15 c \cdot 5}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Combine the numerators over the common denominator.

$19.5 = \frac{102 + 72 c - 15 c \cdot 5}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Multiply $5$ by $- 15$ .

$19.5 = \frac{102 + 72 c - 75 c}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Subtract $75 c$ from $72 c$ .

$19.5 = \frac{102 - 3 c}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Factor $3$ out of $102 - 3 c$ .

Factor $3$ out of $102$ .

$19.5 = \frac{3 (34) - 3 c}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Factor $3$ out of $- 3 c$ .

$19.5 = \frac{3 (34) + 3 (- c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Factor $3$ out of $3 (34) + 3 (- c)$ .

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 54 - 6 b - 8 c$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Replace all occurrences of $b$ in $15 = 54 - 6 b - 8 c$ with $\frac{63}{10} - \frac{6 c}{5}$ .

$15 = 54 - 6 (\frac{63}{10} - \frac{6 c}{5}) - 8 c$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Simplify the right side.

Simplify $54 - 6 (\frac{63}{10} - \frac{6 c}{5}) - 8 c$ .

Simplify each term.

Apply the distributive property.

$15 = 54 - 6 (\frac{63}{10}) - 6 (- \frac{6 c}{5}) - 8 c$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Cancel the common factor of $2$ .

Factor $2$ out of $- 6$ .

$15 = 54 + 2 (- 3) (\frac{63}{10}) - 6 (- \frac{6 c}{5}) - 8 c$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Factor $2$ out of $10$ .

$15 = 54 + 2 \cdot (- 3 \frac{63}{2 \cdot 5}) - 6 (- \frac{6 c}{5}) - 8 c$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Cancel the common factor.

$15 = 54 + 2 \cdot (- 3 \frac{63}{2 \cdot 5}) - 6 (- \frac{6 c}{5}) - 8 c$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Rewrite the expression.

$15 = 54 - 3 (\frac{63}{5}) - 6 (- \frac{6 c}{5}) - 8 c$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$15 = 54 - 3 (\frac{63}{5}) - 6 (- \frac{6 c}{5}) - 8 c$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Combine $- 3$ and $\frac{63}{5}$ .

$15 = 54 + \frac{- 3 \cdot 63}{5} - 6 (- \frac{6 c}{5}) - 8 c$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Multiply $- 3$ by $63$ .

$15 = 54 + \frac{- 189}{5} - 6 (- \frac{6 c}{5}) - 8 c$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Multiply $- 6 (- \frac{6 c}{5})$ .

Multiply $- 1$ by $- 6$ .

$15 = 54 + \frac{- 189}{5} + 6 (\frac{6 c}{5}) - 8 c$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Combine $6$ and $\frac{6 c}{5}$ .

$15 = 54 + \frac{- 189}{5} + \frac{6 (6 c)}{5} - 8 c$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Multiply $6$ by $6$ .

$15 = 54 + \frac{- 189}{5} + \frac{36 c}{5} - 8 c$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$15 = 54 + \frac{- 189}{5} + \frac{36 c}{5} - 8 c$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Move the negative in front of the fraction.

$15 = 54 - \frac{189}{5} + \frac{36 c}{5} - 8 c$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$15 = 54 - \frac{189}{5} + \frac{36 c}{5} - 8 c$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

To write $54$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$15 = 54 \cdot \frac{5}{5} - \frac{189}{5} + \frac{36 c}{5} - 8 c$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Combine $54$ and $\frac{5}{5}$ .

$15 = \frac{54 \cdot 5}{5} - \frac{189}{5} + \frac{36 c}{5} - 8 c$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Combine the numerators over the common denominator.

$15 = \frac{54 \cdot 5 - 189}{5} + \frac{36 c}{5} - 8 c$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Simplify the numerator.

Multiply $54$ by $5$ .

$15 = \frac{270 - 189}{5} + \frac{36 c}{5} - 8 c$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Subtract $189$ from $270$ .

$15 = \frac{81}{5} + \frac{36 c}{5} - 8 c$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$15 = \frac{81}{5} + \frac{36 c}{5} - 8 c$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

To write $- 8 c$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$15 = \frac{81}{5} + \frac{36 c}{5} - 8 c \cdot \frac{5}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Combine $- 8 c$ and $\frac{5}{5}$ .

$15 = \frac{81}{5} + \frac{36 c}{5} + \frac{- 8 c \cdot 5}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Combine the numerators over the common denominator.

$15 = \frac{81}{5} + \frac{36 c - 8 c \cdot 5}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Combine the numerators over the common denominator.

$15 = \frac{81 + 36 c - 8 c \cdot 5}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Multiply $5$ by $- 8$ .

$15 = \frac{81 + 36 c - 40 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Subtract $40 c$ from $36 c$ .

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$10.5 = 24 - 2 b - 3 c$

$a = 6 - b - c$

Replace all occurrences of $b$ in $10.5 = 24 - 2 b - 3 c$ with $\frac{63}{10} - \frac{6 c}{5}$ .

$10.5 = 24 - 2 (\frac{63}{10} - \frac{6 c}{5}) - 3 c$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 6 - b - c$

Simplify the right side.

Simplify $24 - 2 (\frac{63}{10} - \frac{6 c}{5}) - 3 c$ .

Simplify each term.

Apply the distributive property.

$10.5 = 24 - 2 (\frac{63}{10}) - 2 (- \frac{6 c}{5}) - 3 c$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 6 - b - c$

Cancel the common factor of $2$ .

Factor $2$ out of $- 2$ .

$10.5 = 24 + 2 (- 1) (\frac{63}{10}) - 2 (- \frac{6 c}{5}) - 3 c$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 6 - b - c$

Factor $2$ out of $10$ .

$10.5 = 24 + 2 \cdot (- 1 \frac{63}{2 \cdot 5}) - 2 (- \frac{6 c}{5}) - 3 c$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 6 - b - c$

Cancel the common factor.

$10.5 = 24 + 2 \cdot (- 1 \frac{63}{2 \cdot 5}) - 2 (- \frac{6 c}{5}) - 3 c$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 6 - b - c$

Rewrite the expression.

$10.5 = 24 - 1 (\frac{63}{5}) - 2 (- \frac{6 c}{5}) - 3 c$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 6 - b - c$

$10.5 = 24 - 1 (\frac{63}{5}) - 2 (- \frac{6 c}{5}) - 3 c$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 6 - b - c$

Multiply $- 2 (- \frac{6 c}{5})$ .

Multiply $- 1$ by $- 2$ .

$10.5 = 24 - 1 (\frac{63}{5}) + 2 (\frac{6 c}{5}) - 3 c$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 6 - b - c$

Combine $2$ and $\frac{6 c}{5}$ .

$10.5 = 24 - 1 (\frac{63}{5}) + \frac{2 (6 c)}{5} - 3 c$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 6 - b - c$

Multiply $6$ by $2$ .

$10.5 = 24 - 1 (\frac{63}{5}) + \frac{12 c}{5} - 3 c$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 6 - b - c$

$10.5 = 24 - 1 (\frac{63}{5}) + \frac{12 c}{5} - 3 c$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 6 - b - c$

Rewrite $- 1 (\frac{63}{5})$ as $- (\frac{63}{5})$ .

$10.5 = 24 - \frac{63}{5} + \frac{12 c}{5} - 3 c$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 6 - b - c$

$10.5 = 24 - \frac{63}{5} + \frac{12 c}{5} - 3 c$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 6 - b - c$

To write $24$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$10.5 = 24 \cdot \frac{5}{5} - \frac{63}{5} + \frac{12 c}{5} - 3 c$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 6 - b - c$

Combine $24$ and $\frac{5}{5}$ .

$10.5 = \frac{24 \cdot 5}{5} - \frac{63}{5} + \frac{12 c}{5} - 3 c$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 6 - b - c$

Combine the numerators over the common denominator.

$10.5 = \frac{24 \cdot 5 - 63}{5} + \frac{12 c}{5} - 3 c$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 6 - b - c$

Simplify the numerator.

Multiply $24$ by $5$ .

$10.5 = \frac{120 - 63}{5} + \frac{12 c}{5} - 3 c$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 6 - b - c$

Subtract $63$ from $120$ .

$10.5 = \frac{57}{5} + \frac{12 c}{5} - 3 c$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 6 - b - c$

$10.5 = \frac{57}{5} + \frac{12 c}{5} - 3 c$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 6 - b - c$

To write $- 3 c$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$10.5 = \frac{57}{5} + \frac{12 c}{5} - 3 c \cdot \frac{5}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 6 - b - c$

Combine $- 3 c$ and $\frac{5}{5}$ .

$10.5 = \frac{57}{5} + \frac{12 c}{5} + \frac{- 3 c \cdot 5}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 6 - b - c$

Combine the numerators over the common denominator.

$10.5 = \frac{57}{5} + \frac{12 c - 3 c \cdot 5}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 6 - b - c$

Combine the numerators over the common denominator.

$10.5 = \frac{57 + 12 c - 3 c \cdot 5}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 6 - b - c$

Multiply $5$ by $- 3$ .

$10.5 = \frac{57 + 12 c - 15 c}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 6 - b - c$

Subtract $15 c$ from $12 c$ .

$10.5 = \frac{57 - 3 c}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 6 - b - c$

Factor $3$ out of $57 - 3 c$ .

Factor $3$ out of $57$ .

$10.5 = \frac{3 (19) - 3 c}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 6 - b - c$

Factor $3$ out of $- 3 c$ .

$10.5 = \frac{3 (19) + 3 (- c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 6 - b - c$

Factor $3$ out of $3 (19) + 3 (- c)$ .

$10.5 = \frac{3 (19 - c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 6 - b - c$

$10.5 = \frac{3 (19 - c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 6 - b - c$

$10.5 = \frac{3 (19 - c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 6 - b - c$

$10.5 = \frac{3 (19 - c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 6 - b - c$

Replace all occurrences of $b$ in $a = 6 - b - c$ with $\frac{63}{10} - \frac{6 c}{5}$ .

$a = 6 - (\frac{63}{10} - \frac{6 c}{5}) - c$

$10.5 = \frac{3 (19 - c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Simplify the right side.

Simplify $6 - (\frac{63}{10} - \frac{6 c}{5}) - c$ .

Simplify each term.

Apply the distributive property.

$a = 6 - \frac{63}{10} + \frac{6 c}{5} - c$

$10.5 = \frac{3 (19 - c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Multiply $- - \frac{6 c}{5}$ .

Multiply $- 1$ by $- 1$ .

$a = 6 - \frac{63}{10} + 1 (\frac{6 c}{5}) - c$

$10.5 = \frac{3 (19 - c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Multiply $\frac{6 c}{5}$ by $1$ .

$a = 6 - \frac{63}{10} + \frac{6 c}{5} - c$

$10.5 = \frac{3 (19 - c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 6 - \frac{63}{10} + \frac{6 c}{5} - c$

$10.5 = \frac{3 (19 - c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 6 - \frac{63}{10} + \frac{6 c}{5} - c$

$10.5 = \frac{3 (19 - c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

To write $6$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$a = 6 \cdot \frac{10}{10} - \frac{63}{10} + \frac{6 c}{5} - c$

$10.5 = \frac{3 (19 - c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Combine $6$ and $\frac{10}{10}$ .

$a = \frac{6 \cdot 10}{10} - \frac{63}{10} + \frac{6 c}{5} - c$

$10.5 = \frac{3 (19 - c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Combine the numerators over the common denominator.

$a = \frac{6 \cdot 10 - 63}{10} + \frac{6 c}{5} - c$

$10.5 = \frac{3 (19 - c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Simplify the numerator.

Multiply $6$ by $10$ .

$a = \frac{60 - 63}{10} + \frac{6 c}{5} - c$

$10.5 = \frac{3 (19 - c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Subtract $63$ from $60$ .

$a = \frac{- 3}{10} + \frac{6 c}{5} - c$

$10.5 = \frac{3 (19 - c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = \frac{- 3}{10} + \frac{6 c}{5} - c$

$10.5 = \frac{3 (19 - c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Move the negative in front of the fraction.

$a = - \frac{3}{10} + \frac{6 c}{5} - c$

$10.5 = \frac{3 (19 - c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

To write $- c$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$a = - \frac{3}{10} + \frac{6 c}{5} - c \cdot \frac{5}{5}$

$10.5 = \frac{3 (19 - c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Simplify terms.

Combine $- c$ and $\frac{5}{5}$ .

$a = - \frac{3}{10} + \frac{6 c}{5} + \frac{- c \cdot 5}{5}$

$10.5 = \frac{3 (19 - c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Combine the numerators over the common denominator.

$a = - \frac{3}{10} + \frac{6 c - c \cdot 5}{5}$

$10.5 = \frac{3 (19 - c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = - \frac{3}{10} + \frac{6 c - c \cdot 5}{5}$

$10.5 = \frac{3 (19 - c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Simplify each term.

Simplify the numerator.

Factor $c$ out of $6 c - c \cdot 5$ .

Factor $c$ out of $6 c$ .

$a = - \frac{3}{10} + \frac{c \cdot 6 - c \cdot 5}{5}$

$10.5 = \frac{3 (19 - c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Factor $c$ out of $- c \cdot 5$ .

$a = - \frac{3}{10} + \frac{c \cdot 6 + c (- 1 \cdot 5)}{5}$

$10.5 = \frac{3 (19 - c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Factor $c$ out of $c \cdot 6 + c (- 1 \cdot 5)$ .

$a = - \frac{3}{10} + \frac{c (6 - 1 \cdot 5)}{5}$

$10.5 = \frac{3 (19 - c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = - \frac{3}{10} + \frac{c (6 - 1 \cdot 5)}{5}$

$10.5 = \frac{3 (19 - c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Multiply $- 1$ by $5$ .

$a = - \frac{3}{10} + \frac{c (6 - 5)}{5}$

$10.5 = \frac{3 (19 - c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Subtract $5$ from $6$ .

$a = - \frac{3}{10} + \frac{c \cdot 1}{5}$

$10.5 = \frac{3 (19 - c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = - \frac{3}{10} + \frac{c \cdot 1}{5}$

$10.5 = \frac{3 (19 - c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Multiply $c$ by $1$ .

$a = - \frac{3}{10} + \frac{c}{5}$

$10.5 = \frac{3 (19 - c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = - \frac{3}{10} + \frac{c}{5}$

$10.5 = \frac{3 (19 - c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = - \frac{3}{10} + \frac{c}{5}$

$10.5 = \frac{3 (19 - c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = - \frac{3}{10} + \frac{c}{5}$

$10.5 = \frac{3 (19 - c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = - \frac{3}{10} + \frac{c}{5}$

$10.5 = \frac{3 (19 - c)}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Solve for $c$ in $10.5 = \frac{3 (19 - c)}{5}$ .

Rewrite the equation as $\frac{3 (19 - c)}{5} = 10.5$ .

$\frac{3 (19 - c)}{5} = 10.5$

$a = - \frac{3}{10} + \frac{c}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Multiply both sides by $5$ .

$\frac{3 (19 - c)}{5} \cdot 5 = 10.5 \cdot 5$

$a = - \frac{3}{10} + \frac{c}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Simplify.

Simplify the left side.

Simplify $\frac{3 (19 - c)}{5} \cdot 5$ .

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{3 (19 - c)}{5} \cdot 5 = 10.5 \cdot 5$

$a = - \frac{3}{10} + \frac{c}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Rewrite the expression.

$3 (19 - c) = 10.5 \cdot 5$

$a = - \frac{3}{10} + \frac{c}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$3 (19 - c) = 10.5 \cdot 5$

$a = - \frac{3}{10} + \frac{c}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Apply the distributive property.

$3 \cdot 19 + 3 (- c) = 10.5 \cdot 5$

$a = - \frac{3}{10} + \frac{c}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Simplify the expression.

Multiply $3$ by $19$ .

$57 + 3 (- c) = 10.5 \cdot 5$

$a = - \frac{3}{10} + \frac{c}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Multiply $- 1$ by $3$ .

$57 - 3 c = 10.5 \cdot 5$

$a = - \frac{3}{10} + \frac{c}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Reorder $57$ and $- 3 c$ .

$- 3 c + 57 = 10.5 \cdot 5$

$a = - \frac{3}{10} + \frac{c}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$- 3 c + 57 = 10.5 \cdot 5$

$a = - \frac{3}{10} + \frac{c}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$- 3 c + 57 = 10.5 \cdot 5$

$a = - \frac{3}{10} + \frac{c}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$- 3 c + 57 = 10.5 \cdot 5$

$a = - \frac{3}{10} + \frac{c}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Simplify the right side.

Multiply $10.5$ by $5$ .

$- 3 c + 57 = 52.5$

$a = - \frac{3}{10} + \frac{c}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$- 3 c + 57 = 52.5$

$a = - \frac{3}{10} + \frac{c}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$- 3 c + 57 = 52.5$

$a = - \frac{3}{10} + \frac{c}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Solve for $c$ .

Move all terms not containing $c$ to the right side of the equation.

Subtract $57$ from both sides of the equation.

$- 3 c = 52.5 - 57$

$a = - \frac{3}{10} + \frac{c}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Subtract $57$ from $52.5$ .

$- 3 c = - 4.5$

$a = - \frac{3}{10} + \frac{c}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$- 3 c = - 4.5$

$a = - \frac{3}{10} + \frac{c}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Divide each term in $- 3 c = - 4.5$ by $- 3$ and simplify.

Divide each term in $- 3 c = - 4.5$ by $- 3$ .

$\frac{- 3 c}{- 3} = \frac{- 4.5}{- 3}$

$a = - \frac{3}{10} + \frac{c}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 c}{- 3} = \frac{- 4.5}{- 3}$

$a = - \frac{3}{10} + \frac{c}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Divide $c$ by $1$ .

$c = \frac{- 4.5}{- 3}$

$a = - \frac{3}{10} + \frac{c}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$c = \frac{- 4.5}{- 3}$

$a = - \frac{3}{10} + \frac{c}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$c = \frac{- 4.5}{- 3}$

$a = - \frac{3}{10} + \frac{c}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Simplify the right side.

Divide $- 4.5$ by $- 3$ .

$c = 1.5$

$a = - \frac{3}{10} + \frac{c}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$c = 1.5$

$a = - \frac{3}{10} + \frac{c}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$c = 1.5$

$a = - \frac{3}{10} + \frac{c}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$c = 1.5$

$a = - \frac{3}{10} + \frac{c}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$c = 1.5$

$a = - \frac{3}{10} + \frac{c}{5}$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Replace all occurrences of $c$ with $1.5$ in each equation.

Replace all occurrences of $c$ in $a = - \frac{3}{10} + \frac{c}{5}$ with $1.5$ .

$a = - \frac{3}{10} + \frac{1.5}{5}$

$c = 1.5$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Simplify the right side.

Simplify $- \frac{3}{10} + \frac{1.5}{5}$ .

Divide $1.5$ by $5$ .

$a = - \frac{3}{10} + 0.3$

$c = 1.5$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

To write $0.3$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$a = - \frac{3}{10} + 0.3 \cdot \frac{10}{10}$

$c = 1.5$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Combine $0.3$ and $\frac{10}{10}$ .

$a = - \frac{3}{10} + \frac{0.3 \cdot 10}{10}$

$c = 1.5$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Combine the numerators over the common denominator.

$a = \frac{- 3 + 0.3 \cdot 10}{10}$

$c = 1.5$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Simplify the numerator.

Multiply $0.3$ by $10$ .

$a = \frac{- 3 + 3}{10}$

$c = 1.5$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Add $- 3$ and $3$ .

$a = \frac{0}{10}$

$c = 1.5$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = \frac{0}{10}$

$c = 1.5$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Divide $0$ by $10$ .

$a = 0$

$c = 1.5$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 0$

$c = 1.5$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$a = 0$

$c = 1.5$

$15 = \frac{81 - 4 c}{5}$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Replace all occurrences of $c$ in $15 = \frac{81 - 4 c}{5}$ with $1.5$ .

$15 = \frac{81 - 4 \cdot 1.5}{5}$

$a = 0$

$c = 1.5$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Simplify the right side.

Simplify $\frac{81 - 4 \cdot 1.5}{5}$ .

Simplify the numerator.

Multiply $- 4$ by $1.5$ .

$15 = \frac{81 - 6}{5}$

$a = 0$

$c = 1.5$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Subtract $6$ from $81$ .

$15 = \frac{75}{5}$

$a = 0$

$c = 1.5$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = \frac{75}{5}$

$a = 0$

$c = 1.5$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Divide $75$ by $5$ .

$15 = 15$

$a = 0$

$c = 1.5$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 15$

$a = 0$

$c = 1.5$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

$15 = 15$

$a = 0$

$c = 1.5$

$19.5 = \frac{3 (34 - c)}{5}$

$b = \frac{63}{10} - \frac{6 c}{5}$

Replace all occurrences of $c$ in $19.5 = \frac{3 (34 - c)}{5}$ with $1.5$ .

$19.5 = \frac{3 (34 - (1.5))}{5}$

$15 = 15$

$a = 0$

$c = 1.5$

$b = \frac{63}{10} - \frac{6 c}{5}$

Simplify the right side.

Simplify $\frac{3 (34 - (1.5))}{5}$ .

Simplify the numerator.

Multiply $- 1$ by $1.5$ .

$19.5 = \frac{3 (34 - 1.5)}{5}$

$15 = 15$

$a = 0$

$c = 1.5$

$b = \frac{63}{10} - \frac{6 c}{5}$

Subtract $1.5$ from $34$ .

$19.5 = \frac{3 \cdot 32.5}{5}$

$15 = 15$

$a = 0$

$c = 1.5$

$b = \frac{63}{10} - \frac{6 c}{5}$

$19.5 = \frac{3 \cdot 32.5}{5}$

$15 = 15$

$a = 0$

$c = 1.5$

$b = \frac{63}{10} - \frac{6 c}{5}$

Simplify the expression.

Multiply $3$ by $32.5$ .

$19.5 = \frac{97.5}{5}$

$15 = 15$

$a = 0$

$c = 1.5$

$b = \frac{63}{10} - \frac{6 c}{5}$

Divide $97.5$ by $5$ .

$19.5 = 19.5$

$15 = 15$

$a = 0$

$c = 1.5$

$b = \frac{63}{10} - \frac{6 c}{5}$

$19.5 = 19.5$

$15 = 15$

$a = 0$

$c = 1.5$

$b = \frac{63}{10} - \frac{6 c}{5}$

$19.5 = 19.5$

$15 = 15$

$a = 0$

$c = 1.5$

$b = \frac{63}{10} - \frac{6 c}{5}$

$19.5 = 19.5$

$15 = 15$

$a = 0$

$c = 1.5$

$b = \frac{63}{10} - \frac{6 c}{5}$

Replace all occurrences of $c$ in $b = \frac{63}{10} - \frac{6 c}{5}$ with $1.5$ .

$b = \frac{63}{10} - \frac{6 (1.5)}{5}$

$19.5 = 19.5$

$15 = 15$

$a = 0$

$c = 1.5$

Simplify the right side.

Simplify $\frac{63}{10} - \frac{6 (1.5)}{5}$ .

Simplify each term.

Multiply $6$ by $1.5$ .

$b = \frac{63}{10} - \frac{9}{5}$

$19.5 = 19.5$

$15 = 15$

$a = 0$

$c = 1.5$

Cancel the common factor of $9$ and $5$ .

Rewrite $9$ as $1 (9)$ .

$b = \frac{63}{10} - \frac{1 (9)}{5}$

$19.5 = 19.5$

$15 = 15$

$a = 0$

$c = 1.5$

Cancel the common factors.

Rewrite $5$ as $1 (5)$ .

$b = \frac{63}{10} - \frac{1 \cdot 9}{1 \cdot 5}$

$19.5 = 19.5$

$15 = 15$

$a = 0$

$c = 1.5$

Cancel the common factor.

$b = \frac{63}{10} - \frac{1 \cdot 9}{1 \cdot 5}$

$19.5 = 19.5$

$15 = 15$

$a = 0$

$c = 1.5$

Rewrite the expression.

$b = \frac{63}{10} - \frac{9}{5}$

$19.5 = 19.5$

$15 = 15$

$a = 0$

$c = 1.5$

$b = \frac{63}{10} - \frac{9}{5}$

$19.5 = 19.5$

$15 = 15$

$a = 0$

$c = 1.5$

$b = \frac{63}{10} - \frac{9}{5}$

$19.5 = 19.5$

$15 = 15$

$a = 0$

$c = 1.5$

$b = \frac{63}{10} - \frac{9}{5}$

$19.5 = 19.5$

$15 = 15$

$a = 0$

$c = 1.5$

To write $- \frac{9}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$b = \frac{63}{10} - \frac{9}{5} \cdot \frac{2}{2}$

$19.5 = 19.5$

$15 = 15$

$a = 0$

$c = 1.5$

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{9}{5}$ by $\frac{2}{2}$ .

$b = \frac{63}{10} - \frac{9 \cdot 2}{5 \cdot 2}$

$19.5 = 19.5$

$15 = 15$

$a = 0$

$c = 1.5$

Multiply $5$ by $2$ .

$b = \frac{63}{10} - \frac{9 \cdot 2}{10}$

$19.5 = 19.5$

$15 = 15$

$a = 0$

$c = 1.5$

$b = \frac{63}{10} - \frac{9 \cdot 2}{10}$

$19.5 = 19.5$

$15 = 15$

$a = 0$

$c = 1.5$

Combine the numerators over the common denominator.

$b = \frac{63 - 9 \cdot 2}{10}$

$19.5 = 19.5$

$15 = 15$

$a = 0$

$c = 1.5$

Simplify the numerator.

Multiply $- 9$ by $2$ .

$b = \frac{63 - 18}{10}$

$19.5 = 19.5$

$15 = 15$

$a = 0$

$c = 1.5$

Subtract $18$ from $63$ .

$b = \frac{45}{10}$

$19.5 = 19.5$

$15 = 15$

$a = 0$

$c = 1.5$

$b = \frac{45}{10}$

$19.5 = 19.5$

$15 = 15$

$a = 0$

$c = 1.5$

Cancel the common factor of $45$ and $10$ .

Factor $5$ out of $45$ .

$b = \frac{5 (9)}{10}$

$19.5 = 19.5$

$15 = 15$

$a = 0$

$c = 1.5$

Cancel the common factors.

Factor $5$ out of $10$ .

$b = \frac{5 \cdot 9}{5 \cdot 2}$

$19.5 = 19.5$

$15 = 15$

$a = 0$

$c = 1.5$

Cancel the common factor.

$b = \frac{5 \cdot 9}{5 \cdot 2}$

$19.5 = 19.5$

$15 = 15$

$a = 0$

$c = 1.5$

Rewrite the expression.

$b = \frac{9}{2}$

$19.5 = 19.5$

$15 = 15$

$a = 0$

$c = 1.5$

$b = \frac{9}{2}$

$19.5 = 19.5$

$15 = 15$

$a = 0$

$c = 1.5$

$b = \frac{9}{2}$

$19.5 = 19.5$

$15 = 15$

$a = 0$

$c = 1.5$

$b = \frac{9}{2}$

$19.5 = 19.5$

$15 = 15$

$a = 0$

$c = 1.5$

$b = \frac{9}{2}$

$19.5 = 19.5$

$15 = 15$

$a = 0$

$c = 1.5$

$b = \frac{9}{2}$

$19.5 = 19.5$

$15 = 15$

$a = 0$

$c = 1.5$

Remove any equations from the system that are always true.

$b = \frac{9}{2}$

$a = 0$

$c = 1.5$

List all of the solutions.

$b = \frac{9}{2}, a = 0, c = 1.5$

Calculate the value of $y$ using each $x$ value in the table and compare this value to the given $y$ value in the table.

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = 0$ , $b = \frac{9}{2}$ , $c = 1.5$ , and $x = 1$ .

Simplify each term.

One to any power is one.

$y = 0 \cdot 1 + (\frac{9}{2}) \cdot (1) + 1.5$

Multiply $0$ by $1$ .

$y = 0 + (\frac{9}{2}) \cdot (1) + 1.5$

Multiply $\frac{9}{2}$ by $1$ .

$y = 0 + \frac{9}{2} + 1.5$

Find the common denominator.

Write $0$ as a fraction with denominator $1$ .

$y = \frac{0}{1} + \frac{9}{2} + 1.5$

Multiply $\frac{0}{1}$ by $\frac{2}{2}$ .

$y = \frac{0}{1} \cdot \frac{2}{2} + \frac{9}{2} + 1.5$

Multiply $\frac{0}{1}$ by $\frac{2}{2}$ .

$y = \frac{0 \cdot 2}{2} + \frac{9}{2} + 1.5$

Write $1.5$ as a fraction with denominator $1$ .

$y = \frac{0 \cdot 2}{2} + \frac{9}{2} + \frac{1.5}{1}$

Multiply $\frac{1.5}{1}$ by $\frac{2}{2}$ .

$y = \frac{0 \cdot 2}{2} + \frac{9}{2} + \frac{1.5}{1} \cdot \frac{2}{2}$

Multiply $\frac{1.5}{1}$ by $\frac{2}{2}$ .

$y = \frac{0 \cdot 2}{2} + \frac{9}{2} + \frac{1.5 \cdot 2}{2}$

Combine the numerators over the common denominator.

$y = \frac{0 \cdot 2 + 9 + 1.5 \cdot 2}{2}$

Simplify each term.

Multiply $0$ by $2$ .

$y = \frac{0 + 9 + 1.5 \cdot 2}{2}$

Multiply $1.5$ by $2$ .

$y = \frac{0 + 9 + 3}{2}$

Simplify the expression.

Add $0$ and $9$ .

$y = \frac{9 + 3}{2}$

Add $9$ and $3$ .

$y = \frac{12}{2}$

Divide $12$ by $2$ .

$y = 6$

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 1$ . This check does not pass, since $y = 6$ and $y = 6$ . The function rule can\'t be quadratic.

$6 \neq 6$

Since $y \neq y$ for the corresponding $x$ values, the function is not quadratic.

The function is not quadratic

Step 3

Check if the function rule is cubic.

To find if the table follows a function rule, check whether the function rule could follow the form $y = a x^{3} + b x^{2} + c x + d$ .

$y = a x^{3} + b x^{2} + c x + d$

Build a set of $4$ equations from the table such that $y = a x^{3} + b x^{2} + c x + d$ .

Calculate the values of $a$ , $b$ , $c$ , and $d$ .

Solve for $a$ in $6 = a + b + c + d$ .

Rewrite the equation as $a + b + c + d = 6$ .

$a + b + c + d = 6$

$10.5 = a \cdot 2^{3} + b + c + d$

$15 = a \cdot 3^{3} + b + c + d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

Move all terms not containing $a$ to the right side of the equation.

Subtract $b$ from both sides of the equation.

$a + c + d = 6 - b$

$10.5 = a \cdot 2^{3} + b + c + d$

$15 = a \cdot 3^{3} + b + c + d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

Subtract $c$ from both sides of the equation.

$a + d = 6 - b - c$

$10.5 = a \cdot 2^{3} + b + c + d$

$15 = a \cdot 3^{3} + b + c + d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

Subtract $d$ from both sides of the equation.

$a = 6 - b - c - d$

$10.5 = a \cdot 2^{3} + b + c + d$

$15 = a \cdot 3^{3} + b + c + d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

$a = 6 - b - c - d$

$10.5 = a \cdot 2^{3} + b + c + d$

$15 = a \cdot 3^{3} + b + c + d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

$a = 6 - b - c - d$

$10.5 = a \cdot 2^{3} + b + c + d$

$15 = a \cdot 3^{3} + b + c + d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

Replace all occurrences of $a$ with $6 - b - c - d$ in each equation.

Replace all occurrences of $a$ in $10.5 = a \cdot 2^{3} + b + c + d$ with $6 - b - c - d$ .

$10.5 = (6 - b - c - d) \cdot 2^{3} + b + c + d$

$a = 6 - b - c - d$

$15 = a \cdot 3^{3} + b + c + d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

Simplify the right side.

Simplify $(6 - b - c - d) \cdot 2^{3} + b + c + d$ .

Simplify each term.

Raise $2$ to the power of $3$ .

$10.5 = (6 - b - c - d) \cdot 8 + b + c + d$

$a = 6 - b - c - d$

$15 = a \cdot 3^{3} + b + c + d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

Apply the distributive property.

$10.5 = 6 \cdot 8 - b \cdot 8 - c \cdot 8 - d \cdot 8 + b + c + d$

$a = 6 - b - c - d$

$15 = a \cdot 3^{3} + b + c + d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

Simplify.

Multiply $6$ by $8$ .

$10.5 = 48 - b \cdot 8 - c \cdot 8 - d \cdot 8 + b + c + d$

$a = 6 - b - c - d$

$15 = a \cdot 3^{3} + b + c + d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

Multiply $8$ by $- 1$ .

$10.5 = 48 - 8 b - c \cdot 8 - d \cdot 8 + b + c + d$

$a = 6 - b - c - d$

$15 = a \cdot 3^{3} + b + c + d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

Multiply $8$ by $- 1$ .

$10.5 = 48 - 8 b - 8 c - d \cdot 8 + b + c + d$

$a = 6 - b - c - d$

$15 = a \cdot 3^{3} + b + c + d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

Multiply $8$ by $- 1$ .

$10.5 = 48 - 8 b - 8 c - 8 d + b + c + d$

$a = 6 - b - c - d$

$15 = a \cdot 3^{3} + b + c + d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

$10.5 = 48 - 8 b - 8 c - 8 d + b + c + d$

$a = 6 - b - c - d$

$15 = a \cdot 3^{3} + b + c + d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

$10.5 = 48 - 8 b - 8 c - 8 d + b + c + d$

$a = 6 - b - c - d$

$15 = a \cdot 3^{3} + b + c + d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

Simplify by adding terms.

Add $- 8 b$ and $b$ .

$10.5 = 48 - 7 b - 8 c - 8 d + c + d$

$a = 6 - b - c - d$

$15 = a \cdot 3^{3} + b + c + d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

Add $- 8 c$ and $c$ .

$10.5 = 48 - 7 b - 7 c - 8 d + d$

$a = 6 - b - c - d$

$15 = a \cdot 3^{3} + b + c + d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

Add $- 8 d$ and $d$ .

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$15 = a \cdot 3^{3} + b + c + d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$15 = a \cdot 3^{3} + b + c + d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$15 = a \cdot 3^{3} + b + c + d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$15 = a \cdot 3^{3} + b + c + d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

Replace all occurrences of $a$ in $15 = a \cdot 3^{3} + b + c + d$ with $6 - b - c - d$ .

$15 = (6 - b - c - d) \cdot 3^{3} + b + c + d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

Simplify the right side.

Simplify $(6 - b - c - d) \cdot 3^{3} + b + c + d$ .

Simplify each term.

Raise $3$ to the power of $3$ .

$15 = (6 - b - c - d) \cdot 27 + b + c + d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

Apply the distributive property.

$15 = 6 \cdot 27 - b \cdot 27 - c \cdot 27 - d \cdot 27 + b + c + d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

Simplify.

Multiply $6$ by $27$ .

$15 = 162 - b \cdot 27 - c \cdot 27 - d \cdot 27 + b + c + d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

Multiply $27$ by $- 1$ .

$15 = 162 - 27 b - c \cdot 27 - d \cdot 27 + b + c + d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

Multiply $27$ by $- 1$ .

$15 = 162 - 27 b - 27 c - d \cdot 27 + b + c + d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

Multiply $27$ by $- 1$ .

$15 = 162 - 27 b - 27 c - 27 d + b + c + d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

$15 = 162 - 27 b - 27 c - 27 d + b + c + d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

$15 = 162 - 27 b - 27 c - 27 d + b + c + d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

Simplify by adding terms.

Add $- 27 b$ and $b$ .

$15 = 162 - 26 b - 27 c - 27 d + c + d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

Add $- 27 c$ and $c$ .

$15 = 162 - 26 b - 26 c - 27 d + d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

Add $- 27 d$ and $d$ .

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$19.5 = a \cdot 4^{3} + b + c + d$

$24 = a \cdot 5^{3} + b + c + d$

Replace all occurrences of $a$ in $19.5 = a \cdot 4^{3} + b + c + d$ with $6 - b - c - d$ .

$19.5 = (6 - b - c - d) \cdot 4^{3} + b + c + d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$24 = a \cdot 5^{3} + b + c + d$

Simplify the right side.

Simplify $(6 - b - c - d) \cdot 4^{3} + b + c + d$ .

Simplify each term.

Raise $4$ to the power of $3$ .

$19.5 = (6 - b - c - d) \cdot 64 + b + c + d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$24 = a \cdot 5^{3} + b + c + d$

Apply the distributive property.

$19.5 = 6 \cdot 64 - b \cdot 64 - c \cdot 64 - d \cdot 64 + b + c + d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$24 = a \cdot 5^{3} + b + c + d$

Simplify.

Multiply $6$ by $64$ .

$19.5 = 384 - b \cdot 64 - c \cdot 64 - d \cdot 64 + b + c + d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$24 = a \cdot 5^{3} + b + c + d$

Multiply $64$ by $- 1$ .

$19.5 = 384 - 64 b - c \cdot 64 - d \cdot 64 + b + c + d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$24 = a \cdot 5^{3} + b + c + d$

Multiply $64$ by $- 1$ .

$19.5 = 384 - 64 b - 64 c - d \cdot 64 + b + c + d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$24 = a \cdot 5^{3} + b + c + d$

Multiply $64$ by $- 1$ .

$19.5 = 384 - 64 b - 64 c - 64 d + b + c + d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$24 = a \cdot 5^{3} + b + c + d$

$19.5 = 384 - 64 b - 64 c - 64 d + b + c + d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$24 = a \cdot 5^{3} + b + c + d$

$19.5 = 384 - 64 b - 64 c - 64 d + b + c + d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$24 = a \cdot 5^{3} + b + c + d$

Simplify by adding terms.

Add $- 64 b$ and $b$ .

$19.5 = 384 - 63 b - 64 c - 64 d + c + d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$24 = a \cdot 5^{3} + b + c + d$

Add $- 64 c$ and $c$ .

$19.5 = 384 - 63 b - 63 c - 64 d + d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$24 = a \cdot 5^{3} + b + c + d$

Add $- 64 d$ and $d$ .

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$24 = a \cdot 5^{3} + b + c + d$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$24 = a \cdot 5^{3} + b + c + d$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$24 = a \cdot 5^{3} + b + c + d$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$24 = a \cdot 5^{3} + b + c + d$

Replace all occurrences of $a$ in $24 = a \cdot 5^{3} + b + c + d$ with $6 - b - c - d$ .

$24 = (6 - b - c - d) \cdot 5^{3} + b + c + d$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Simplify the right side.

Simplify $(6 - b - c - d) \cdot 5^{3} + b + c + d$ .

Simplify each term.

Raise $5$ to the power of $3$ .

$24 = (6 - b - c - d) \cdot 125 + b + c + d$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Apply the distributive property.

$24 = 6 \cdot 125 - b \cdot 125 - c \cdot 125 - d \cdot 125 + b + c + d$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Simplify.

Multiply $6$ by $125$ .

$24 = 750 - b \cdot 125 - c \cdot 125 - d \cdot 125 + b + c + d$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Multiply $125$ by $- 1$ .

$24 = 750 - 125 b - c \cdot 125 - d \cdot 125 + b + c + d$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Multiply $125$ by $- 1$ .

$24 = 750 - 125 b - 125 c - d \cdot 125 + b + c + d$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Multiply $125$ by $- 1$ .

$24 = 750 - 125 b - 125 c - 125 d + b + c + d$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$24 = 750 - 125 b - 125 c - 125 d + b + c + d$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$24 = 750 - 125 b - 125 c - 125 d + b + c + d$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Simplify by adding terms.

Add $- 125 b$ and $b$ .

$24 = 750 - 124 b - 125 c - 125 d + c + d$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Add $- 125 c$ and $c$ .

$24 = 750 - 124 b - 124 c - 125 d + d$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Add $- 125 d$ and $d$ .

$24 = 750 - 124 b - 124 c - 124 d$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$24 = 750 - 124 b - 124 c - 124 d$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$24 = 750 - 124 b - 124 c - 124 d$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$24 = 750 - 124 b - 124 c - 124 d$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$24 = 750 - 124 b - 124 c - 124 d$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Solve for $b$ in $24 = 750 - 124 b - 124 c - 124 d$ .

Rewrite the equation as $750 - 124 b - 124 c - 124 d = 24$ .

$750 - 124 b - 124 c - 124 d = 24$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Move all terms not containing $b$ to the right side of the equation.

Subtract $750$ from both sides of the equation.

$- 124 b - 124 c - 124 d = 24 - 750$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Add $124 c$ to both sides of the equation.

$- 124 b - 124 d = 24 - 750 + 124 c$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Add $124 d$ to both sides of the equation.

$- 124 b = 24 - 750 + 124 c + 124 d$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Subtract $750$ from $24$ .

$- 124 b = - 726 + 124 c + 124 d$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$- 124 b = - 726 + 124 c + 124 d$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Divide each term in $- 124 b = - 726 + 124 c + 124 d$ by $- 124$ and simplify.

Divide each term in $- 124 b = - 726 + 124 c + 124 d$ by $- 124$ .

$\frac{- 124 b}{- 124} = \frac{- 726}{- 124} + \frac{124 c}{- 124} + \frac{124 d}{- 124}$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Simplify the left side.

Cancel the common factor of $- 124$ .

Cancel the common factor.

$\frac{- 124 b}{- 124} = \frac{- 726}{- 124} + \frac{124 c}{- 124} + \frac{124 d}{- 124}$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Divide $b$ by $1$ .

$b = \frac{- 726}{- 124} + \frac{124 c}{- 124} + \frac{124 d}{- 124}$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$b = \frac{- 726}{- 124} + \frac{124 c}{- 124} + \frac{124 d}{- 124}$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$b = \frac{- 726}{- 124} + \frac{124 c}{- 124} + \frac{124 d}{- 124}$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 726$ and $- 124$ .

Factor $- 2$ out of $- 726$ .

$b = \frac{- 2 \cdot 363}{- 124} + \frac{124 c}{- 124} + \frac{124 d}{- 124}$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Cancel the common factors.

Factor $- 2$ out of $- 124$ .

$b = \frac{- 2 \cdot 363}{- 2 \cdot 62} + \frac{124 c}{- 124} + \frac{124 d}{- 124}$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Cancel the common factor.

$b = \frac{- 2 \cdot 363}{- 2 \cdot 62} + \frac{124 c}{- 124} + \frac{124 d}{- 124}$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Rewrite the expression.

$b = \frac{363}{62} + \frac{124 c}{- 124} + \frac{124 d}{- 124}$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$b = \frac{363}{62} + \frac{124 c}{- 124} + \frac{124 d}{- 124}$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$b = \frac{363}{62} + \frac{124 c}{- 124} + \frac{124 d}{- 124}$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Cancel the common factor of $124$ and $- 124$ .

Factor $124$ out of $124 c$ .

$b = \frac{363}{62} + \frac{124 (c)}{- 124} + \frac{124 d}{- 124}$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Move the negative one from the denominator of $\frac{c}{- 1}$ .

$b = \frac{363}{62} - 1 \cdot c + \frac{124 d}{- 124}$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$b = \frac{363}{62} - 1 \cdot c + \frac{124 d}{- 124}$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Rewrite $- 1 \cdot c$ as $- c$ .

$b = \frac{363}{62} - c + \frac{124 d}{- 124}$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Cancel the common factor of $124$ and $- 124$ .

Factor $124$ out of $124 d$ .

$b = \frac{363}{62} - c + \frac{124 (d)}{- 124}$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Move the negative one from the denominator of $\frac{d}{- 1}$ .

$b = \frac{363}{62} - c - 1 \cdot d$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$b = \frac{363}{62} - c - 1 \cdot d$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Rewrite $- 1 \cdot d$ as $- d$ .

$b = \frac{363}{62} - c - d$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$b = \frac{363}{62} - c - d$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$b = \frac{363}{62} - c - d$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$b = \frac{363}{62} - c - d$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$b = \frac{363}{62} - c - d$

$19.5 = 384 - 63 b - 63 c - 63 d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Replace all occurrences of $b$ with $\frac{363}{62} - c - d$ in each equation.

Replace all occurrences of $b$ in $19.5 = 384 - 63 b - 63 c - 63 d$ with $\frac{363}{62} - c - d$ .

$19.5 = 384 - 63 (\frac{363}{62} - c - d) - 63 c - 63 d$

$b = \frac{363}{62} - c - d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Simplify the right side.

Simplify $384 - 63 (\frac{363}{62} - c - d) - 63 c - 63 d$ .

Simplify each term.

Apply the distributive property.

$19.5 = 384 - 63 (\frac{363}{62}) - 63 (- c) - 63 (- d) - 63 c - 63 d$

$b = \frac{363}{62} - c - d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Simplify.

Multiply $- 63 (\frac{363}{62})$ .

Combine $- 63$ and $\frac{363}{62}$ .

$19.5 = 384 + \frac{- 63 \cdot 363}{62} - 63 (- c) - 63 (- d) - 63 c - 63 d$

$b = \frac{363}{62} - c - d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Multiply $- 63$ by $363$ .

$19.5 = 384 + \frac{- 22869}{62} - 63 (- c) - 63 (- d) - 63 c - 63 d$

$b = \frac{363}{62} - c - d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$19.5 = 384 + \frac{- 22869}{62} - 63 (- c) - 63 (- d) - 63 c - 63 d$

$b = \frac{363}{62} - c - d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Multiply $- 1$ by $- 63$ .

$19.5 = 384 + \frac{- 22869}{62} + 63 c - 63 (- d) - 63 c - 63 d$

$b = \frac{363}{62} - c - d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Multiply $- 1$ by $- 63$ .

$19.5 = 384 + \frac{- 22869}{62} + 63 c + 63 d - 63 c - 63 d$

$b = \frac{363}{62} - c - d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$19.5 = 384 + \frac{- 22869}{62} + 63 c + 63 d - 63 c - 63 d$

$b = \frac{363}{62} - c - d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Move the negative in front of the fraction.

$19.5 = 384 - \frac{22869}{62} + 63 c + 63 d - 63 c - 63 d$

$b = \frac{363}{62} - c - d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$19.5 = 384 - \frac{22869}{62} + 63 c + 63 d - 63 c - 63 d$

$b = \frac{363}{62} - c - d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Combine the opposite terms in $384 - \frac{22869}{62} + 63 c + 63 d - 63 c - 63 d$ .

Subtract $63 c$ from $63 c$ .

$19.5 = 384 - \frac{22869}{62} + 63 d + 0 - 63 d$

$b = \frac{363}{62} - c - d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Add $384 - \frac{22869}{62} + 63 d$ and $0$ .

$19.5 = 384 - \frac{22869}{62} + 63 d - 63 d$

$b = \frac{363}{62} - c - d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Subtract $63 d$ from $63 d$ .

$19.5 = 384 - \frac{22869}{62} + 0$

$b = \frac{363}{62} - c - d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Add $384 - \frac{22869}{62}$ and $0$ .

$19.5 = 384 - \frac{22869}{62}$

$b = \frac{363}{62} - c - d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$19.5 = 384 - \frac{22869}{62}$

$b = \frac{363}{62} - c - d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

To write $384$ as a fraction with a common denominator, multiply by $\frac{62}{62}$ .

$19.5 = 384 \cdot \frac{62}{62} - \frac{22869}{62}$

$b = \frac{363}{62} - c - d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Combine $384$ and $\frac{62}{62}$ .

$19.5 = \frac{384 \cdot 62}{62} - \frac{22869}{62}$

$b = \frac{363}{62} - c - d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Combine the numerators over the common denominator.

$19.5 = \frac{384 \cdot 62 - 22869}{62}$

$b = \frac{363}{62} - c - d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Simplify the numerator.

Multiply $384$ by $62$ .

$19.5 = \frac{23808 - 22869}{62}$

$b = \frac{363}{62} - c - d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Subtract $22869$ from $23808$ .

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$15 = 162 - 26 b - 26 c - 26 d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Replace all occurrences of $b$ in $15 = 162 - 26 b - 26 c - 26 d$ with $\frac{363}{62} - c - d$ .

$15 = 162 - 26 (\frac{363}{62} - c - d) - 26 c - 26 d$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Simplify the right side.

Simplify $162 - 26 (\frac{363}{62} - c - d) - 26 c - 26 d$ .

Simplify each term.

Apply the distributive property.

$15 = 162 - 26 (\frac{363}{62}) - 26 (- c) - 26 (- d) - 26 c - 26 d$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Simplify.

Cancel the common factor of $2$ .

Factor $2$ out of $- 26$ .

$15 = 162 + 2 (- 13) (\frac{363}{62}) - 26 (- c) - 26 (- d) - 26 c - 26 d$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Factor $2$ out of $62$ .

$15 = 162 + 2 \cdot (- 13 \frac{363}{2 \cdot 31}) - 26 (- c) - 26 (- d) - 26 c - 26 d$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Cancel the common factor.

$15 = 162 + 2 \cdot (- 13 \frac{363}{2 \cdot 31}) - 26 (- c) - 26 (- d) - 26 c - 26 d$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Rewrite the expression.

$15 = 162 - 13 (\frac{363}{31}) - 26 (- c) - 26 (- d) - 26 c - 26 d$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$15 = 162 - 13 (\frac{363}{31}) - 26 (- c) - 26 (- d) - 26 c - 26 d$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Combine $- 13$ and $\frac{363}{31}$ .

$15 = 162 + \frac{- 13 \cdot 363}{31} - 26 (- c) - 26 (- d) - 26 c - 26 d$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Multiply $- 13$ by $363$ .

$15 = 162 + \frac{- 4719}{31} - 26 (- c) - 26 (- d) - 26 c - 26 d$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Multiply $- 1$ by $- 26$ .

$15 = 162 + \frac{- 4719}{31} + 26 c - 26 (- d) - 26 c - 26 d$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Multiply $- 1$ by $- 26$ .

$15 = 162 + \frac{- 4719}{31} + 26 c + 26 d - 26 c - 26 d$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$15 = 162 + \frac{- 4719}{31} + 26 c + 26 d - 26 c - 26 d$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Move the negative in front of the fraction.

$15 = 162 - \frac{4719}{31} + 26 c + 26 d - 26 c - 26 d$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$15 = 162 - \frac{4719}{31} + 26 c + 26 d - 26 c - 26 d$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Combine the opposite terms in $162 - \frac{4719}{31} + 26 c + 26 d - 26 c - 26 d$ .

Subtract $26 c$ from $26 c$ .

$15 = 162 - \frac{4719}{31} + 26 d + 0 - 26 d$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Add $162 - \frac{4719}{31} + 26 d$ and $0$ .

$15 = 162 - \frac{4719}{31} + 26 d - 26 d$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Subtract $26 d$ from $26 d$ .

$15 = 162 - \frac{4719}{31} + 0$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Add $162 - \frac{4719}{31}$ and $0$ .

$15 = 162 - \frac{4719}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$15 = 162 - \frac{4719}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

To write $162$ as a fraction with a common denominator, multiply by $\frac{31}{31}$ .

$15 = 162 \cdot \frac{31}{31} - \frac{4719}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Combine $162$ and $\frac{31}{31}$ .

$15 = \frac{162 \cdot 31}{31} - \frac{4719}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Combine the numerators over the common denominator.

$15 = \frac{162 \cdot 31 - 4719}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Simplify the numerator.

Multiply $162$ by $31$ .

$15 = \frac{5022 - 4719}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Subtract $4719$ from $5022$ .

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$10.5 = 48 - 7 b - 7 c - 7 d$

$a = 6 - b - c - d$

Replace all occurrences of $b$ in $10.5 = 48 - 7 b - 7 c - 7 d$ with $\frac{363}{62} - c - d$ .

$10.5 = 48 - 7 (\frac{363}{62} - c - d) - 7 c - 7 d$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$a = 6 - b - c - d$

Simplify the right side.

Simplify $48 - 7 (\frac{363}{62} - c - d) - 7 c - 7 d$ .

Simplify each term.

Apply the distributive property.

$10.5 = 48 - 7 (\frac{363}{62}) - 7 (- c) - 7 (- d) - 7 c - 7 d$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$a = 6 - b - c - d$

Simplify.

Multiply $- 7 (\frac{363}{62})$ .

Combine $- 7$ and $\frac{363}{62}$ .

$10.5 = 48 + \frac{- 7 \cdot 363}{62} - 7 (- c) - 7 (- d) - 7 c - 7 d$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$a = 6 - b - c - d$

Multiply $- 7$ by $363$ .

$10.5 = 48 + \frac{- 2541}{62} - 7 (- c) - 7 (- d) - 7 c - 7 d$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$a = 6 - b - c - d$

$10.5 = 48 + \frac{- 2541}{62} - 7 (- c) - 7 (- d) - 7 c - 7 d$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$a = 6 - b - c - d$

Multiply $- 1$ by $- 7$ .

$10.5 = 48 + \frac{- 2541}{62} + 7 c - 7 (- d) - 7 c - 7 d$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$a = 6 - b - c - d$

Multiply $- 1$ by $- 7$ .

$10.5 = 48 + \frac{- 2541}{62} + 7 c + 7 d - 7 c - 7 d$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$a = 6 - b - c - d$

$10.5 = 48 + \frac{- 2541}{62} + 7 c + 7 d - 7 c - 7 d$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$a = 6 - b - c - d$

Move the negative in front of the fraction.

$10.5 = 48 - \frac{2541}{62} + 7 c + 7 d - 7 c - 7 d$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$a = 6 - b - c - d$

$10.5 = 48 - \frac{2541}{62} + 7 c + 7 d - 7 c - 7 d$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$a = 6 - b - c - d$

Combine the opposite terms in $48 - \frac{2541}{62} + 7 c + 7 d - 7 c - 7 d$ .

Subtract $7 c$ from $7 c$ .

$10.5 = 48 - \frac{2541}{62} + 7 d + 0 - 7 d$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$a = 6 - b - c - d$

Add $48 - \frac{2541}{62} + 7 d$ and $0$ .

$10.5 = 48 - \frac{2541}{62} + 7 d - 7 d$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$a = 6 - b - c - d$

Subtract $7 d$ from $7 d$ .

$10.5 = 48 - \frac{2541}{62} + 0$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$a = 6 - b - c - d$

Add $48 - \frac{2541}{62}$ and $0$ .

$10.5 = 48 - \frac{2541}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$a = 6 - b - c - d$

$10.5 = 48 - \frac{2541}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$a = 6 - b - c - d$

To write $48$ as a fraction with a common denominator, multiply by $\frac{62}{62}$ .

$10.5 = 48 \cdot \frac{62}{62} - \frac{2541}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$a = 6 - b - c - d$

Combine $48$ and $\frac{62}{62}$ .

$10.5 = \frac{48 \cdot 62}{62} - \frac{2541}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$a = 6 - b - c - d$

Combine the numerators over the common denominator.

$10.5 = \frac{48 \cdot 62 - 2541}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$a = 6 - b - c - d$

Simplify the numerator.

Multiply $48$ by $62$ .

$10.5 = \frac{2976 - 2541}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$a = 6 - b - c - d$

Subtract $2541$ from $2976$ .

$10.5 = \frac{435}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$a = 6 - b - c - d$

$10.5 = \frac{435}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$a = 6 - b - c - d$

$10.5 = \frac{435}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$a = 6 - b - c - d$

$10.5 = \frac{435}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$a = 6 - b - c - d$

Replace all occurrences of $b$ in $a = 6 - b - c - d$ with $\frac{363}{62} - c - d$ .

$a = 6 - (\frac{363}{62} - c - d) - c - d$

$10.5 = \frac{435}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

Simplify the right side.

Simplify $6 - (\frac{363}{62} - c - d) - c - d$ .

Simplify each term.

Apply the distributive property.

$a = 6 - \frac{363}{62} + c + d - c - d$

$10.5 = \frac{435}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

Simplify.

Multiply $- - c$ .

Multiply $- 1$ by $- 1$ .

$a = 6 - \frac{363}{62} + 1 c + d - c - d$

$10.5 = \frac{435}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

Multiply $c$ by $1$ .

$a = 6 - \frac{363}{62} + c + d - c - d$

$10.5 = \frac{435}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$a = 6 - \frac{363}{62} + c + d - c - d$

$10.5 = \frac{435}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

Multiply $- - d$ .

Multiply $- 1$ by $- 1$ .

$a = 6 - \frac{363}{62} + c + 1 d - c - d$

$10.5 = \frac{435}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

Multiply $d$ by $1$ .

$a = 6 - \frac{363}{62} + c + d - c - d$

$10.5 = \frac{435}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$a = 6 - \frac{363}{62} + c + d - c - d$

$10.5 = \frac{435}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$a = 6 - \frac{363}{62} + c + d - c - d$

$10.5 = \frac{435}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$a = 6 - \frac{363}{62} + c + d - c - d$

$10.5 = \frac{435}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

Combine the opposite terms in $6 - \frac{363}{62} + c + d - c - d$ .

Subtract $c$ from $c$ .

$a = 6 - \frac{363}{62} + d + 0 - d$

$10.5 = \frac{435}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

Add $6 - \frac{363}{62} + d$ and $0$ .

$a = 6 - \frac{363}{62} + d - d$

$10.5 = \frac{435}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

Subtract $d$ from $d$ .

$a = 6 - \frac{363}{62} + 0$

$10.5 = \frac{435}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

Add $6 - \frac{363}{62}$ and $0$ .

$a = 6 - \frac{363}{62}$

$10.5 = \frac{435}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$a = 6 - \frac{363}{62}$

$10.5 = \frac{435}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

To write $6$ as a fraction with a common denominator, multiply by $\frac{62}{62}$ .

$a = 6 \cdot \frac{62}{62} - \frac{363}{62}$

$10.5 = \frac{435}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

Combine $6$ and $\frac{62}{62}$ .

$a = \frac{6 \cdot 62}{62} - \frac{363}{62}$

$10.5 = \frac{435}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

Combine the numerators over the common denominator.

$a = \frac{6 \cdot 62 - 363}{62}$

$10.5 = \frac{435}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

Simplify the numerator.

Multiply $6$ by $62$ .

$a = \frac{372 - 363}{62}$

$10.5 = \frac{435}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

Subtract $363$ from $372$ .

$a = \frac{9}{62}$

$10.5 = \frac{435}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$a = \frac{9}{62}$

$10.5 = \frac{435}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$a = \frac{9}{62}$

$10.5 = \frac{435}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$a = \frac{9}{62}$

$10.5 = \frac{435}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

$a = \frac{9}{62}$

$10.5 = \frac{435}{62}$

$15 = \frac{303}{31}$

$19.5 = \frac{939}{62}$

$b = \frac{363}{62} - c - d$

Since $10.5 = \frac{435}{62}$ is not true, there is no solution.

No solution

Calculate the value of $y$ using each $x$ value in the table and compare this value to the given $y$ value in the table.

Calculate the value of $y$ such that $y = a x^{3} + b$ when $a = 0$ , $b = 0$ , $c = 0$ , $d = 0$ , and $x = 1$ .

Simplify each term.

One to any power is one.

$y = 0 \cdot 1 + (0) \cdot (1^{2}) + (0) \cdot ((1) \cdot 0)$

Multiply $0$ by $1$ .

$y = 0 + (0) \cdot (1^{2}) + (0) \cdot ((1) \cdot 0)$

One to any power is one.

$y = 0 + 0 \cdot 1 + (0) \cdot ((1) \cdot 0)$

Multiply $0$ by $1$ .

$y = 0 + 0 + (0) \cdot ((1) \cdot 0)$

Multiply $(0) ((1) \cdot 0)$ .

Multiply $0$ by $1$ .

$y = 0 + 0 + 0 \cdot 0$

Multiply $0$ by $0$ .

$y = 0 + 0 + 0$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 0$

Add $0$ and $0$ .

$y = 0$

If the table has a cubic function rule, $y = y$ for the corresponding $x$ value, $x = 1$ . This check does not pass, since $y = 0$ and $y = 6$ . The function rule can\'t be cubic.

$0 \neq 6$

Since $y \neq y$ for the corresponding $x$ values, the function is not cubic.

The function is not cubic

Step 4

There are no values for $a$ , $b$ , $c$ , and $d$ in the equations $y = a x + b$ , $y = a x^{2} + b x + c$ , and $y = a x^{3} + b x^{2} + c x + d$ that work for every pair of $x$ and $y$ .

The table does not have a function rule that is linear, quadratic, or cubic.

Find the Function Rule table[[x,y],[1,6],[2,10.5],[3,15],[4,19.5],[5,24]]

Related Questions

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