Find the Function Rule table[[x,f(x)],[8,3],[16,4],[32,5]]

$\begin{array}{c} x & f (x) \\ 8 & 3 \\ 16 & 4 \\ 32 & 5 \end{array}$

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 $f (x) = a x + b$ .

$3 = a (8) + b 4 = a (16) + b 5 = a (32) + b$

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

Solve for $b$ in $3 = a (8) + b$ .

Rewrite the equation as $a (8) + b = 3$ .

$a (8) + b = 3$

$4 = a (16) + b$

$5 = a (32) + b$

Move $8$ to the left of $a$ .

$8 a + b = 3$

$4 = a (16) + b$

$5 = a (32) + b$

Subtract $8 a$ from both sides of the equation.

$b = 3 - 8 a$

$4 = a (16) + b$

$5 = a (32) + b$

$b = 3 - 8 a$

$4 = a (16) + b$

$5 = a (32) + b$

Replace all occurrences of $b$ with $3 - 8 a$ in each equation.

Replace all occurrences of $b$ in $4 = a (16) + b$ with $3 - 8 a$ .

$4 = a (16) + 3 - 8 a$

$b = 3 - 8 a$

$5 = a (32) + b$

Simplify $4 = a (16) + 3 - 8 a$ .

Simplify the left side.

Remove parentheses.

$4 = a (16) + 3 - 8 a$

$b = 3 - 8 a$

$5 = a (32) + b$

$4 = a (16) + 3 - 8 a$

$b = 3 - 8 a$

$5 = a (32) + b$

Simplify the right side.

Simplify $a (16) + 3 - 8 a$ .

Move $16$ to the left of $a$ .

$4 = 16 a + 3 - 8 a$

$b = 3 - 8 a$

$5 = a (32) + b$

Subtract $8 a$ from $16 a$ .

$4 = 8 a + 3$

$b = 3 - 8 a$

$5 = a (32) + b$

$4 = 8 a + 3$

$b = 3 - 8 a$

$5 = a (32) + b$

$4 = 8 a + 3$

$b = 3 - 8 a$

$5 = a (32) + b$

$4 = 8 a + 3$

$b = 3 - 8 a$

$5 = a (32) + b$

Replace all occurrences of $b$ in $5 = a (32) + b$ with $3 - 8 a$ .

$5 = a (32) + 3 - 8 a$

$4 = 8 a + 3$

$b = 3 - 8 a$

Simplify $5 = a (32) + 3 - 8 a$ .

Simplify the left side.

Remove parentheses.

$5 = a (32) + 3 - 8 a$

$4 = 8 a + 3$

$b = 3 - 8 a$

$5 = a (32) + 3 - 8 a$

$4 = 8 a + 3$

$b = 3 - 8 a$

Simplify the right side.

Simplify $a (32) + 3 - 8 a$ .

Move $32$ to the left of $a$ .

$5 = 32 a + 3 - 8 a$

$4 = 8 a + 3$

$b = 3 - 8 a$

Subtract $8 a$ from $32 a$ .

$5 = 24 a + 3$

$4 = 8 a + 3$

$b = 3 - 8 a$

$5 = 24 a + 3$

$4 = 8 a + 3$

$b = 3 - 8 a$

$5 = 24 a + 3$

$4 = 8 a + 3$

$b = 3 - 8 a$

$5 = 24 a + 3$

$4 = 8 a + 3$

$b = 3 - 8 a$

$5 = 24 a + 3$

$4 = 8 a + 3$

$b = 3 - 8 a$

Solve for $a$ in $5 = 24 a + 3$ .

Rewrite the equation as $24 a + 3 = 5$ .

$24 a + 3 = 5$

$4 = 8 a + 3$

$b = 3 - 8 a$

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

Subtract $3$ from both sides of the equation.

$24 a = 5 - 3$

$4 = 8 a + 3$

$b = 3 - 8 a$

Subtract $3$ from $5$ .

$24 a = 2$

$4 = 8 a + 3$

$b = 3 - 8 a$

$24 a = 2$

$4 = 8 a + 3$

$b = 3 - 8 a$

Divide each term in $24 a = 2$ by $24$ and simplify.

Divide each term in $24 a = 2$ by $24$ .

$\frac{24 a}{24} = \frac{2}{24}$

$4 = 8 a + 3$

$b = 3 - 8 a$

Simplify the left side.

Cancel the common factor of $24$ .

Cancel the common factor.

$\frac{24 a}{24} = \frac{2}{24}$

$4 = 8 a + 3$

$b = 3 - 8 a$

Divide $a$ by $1$ .

$a = \frac{2}{24}$

$4 = 8 a + 3$

$b = 3 - 8 a$

$a = \frac{2}{24}$

$4 = 8 a + 3$

$b = 3 - 8 a$

$a = \frac{2}{24}$

$4 = 8 a + 3$

$b = 3 - 8 a$

Simplify the right side.

Cancel the common factor of $2$ and $24$ .

Factor $2$ out of $2$ .

$a = \frac{2 (1)}{24}$

$4 = 8 a + 3$

$b = 3 - 8 a$

Cancel the common factors.

Factor $2$ out of $24$ .

$a = \frac{2 \cdot 1}{2 \cdot 12}$

$4 = 8 a + 3$

$b = 3 - 8 a$

Cancel the common factor.

$a = \frac{2 \cdot 1}{2 \cdot 12}$

$4 = 8 a + 3$

$b = 3 - 8 a$

Rewrite the expression.

$a = \frac{1}{12}$

$4 = 8 a + 3$

$b = 3 - 8 a$

$a = \frac{1}{12}$

$4 = 8 a + 3$

$b = 3 - 8 a$

$a = \frac{1}{12}$

$4 = 8 a + 3$

$b = 3 - 8 a$

$a = \frac{1}{12}$

$4 = 8 a + 3$

$b = 3 - 8 a$

$a = \frac{1}{12}$

$4 = 8 a + 3$

$b = 3 - 8 a$

$a = \frac{1}{12}$

$4 = 8 a + 3$

$b = 3 - 8 a$

Replace all occurrences of $a$ with $\frac{1}{12}$ in each equation.

Replace all occurrences of $a$ in $4 = 8 a + 3$ with $\frac{1}{12}$ .

$4 = 8 (\frac{1}{12}) + 3$

$a = \frac{1}{12}$

$b = 3 - 8 a$

Simplify the right side.

Simplify $8 (\frac{1}{12}) + 3$ .

Simplify each term.

Cancel the common factor of $4$ .

Factor $4$ out of $8$ .

$4 = 4 (2) (\frac{1}{12}) + 3$

$a = \frac{1}{12}$

$b = 3 - 8 a$

Factor $4$ out of $12$ .

$4 = 4 \cdot (2 (\frac{1}{4 \cdot 3})) + 3$

$a = \frac{1}{12}$

$b = 3 - 8 a$

Cancel the common factor.

$4 = 4 \cdot (2 (\frac{1}{4 \cdot 3})) + 3$

$a = \frac{1}{12}$

$b = 3 - 8 a$

Rewrite the expression.

$4 = 2 (\frac{1}{3}) + 3$

$a = \frac{1}{12}$

$b = 3 - 8 a$

$4 = 2 (\frac{1}{3}) + 3$

$a = \frac{1}{12}$

$b = 3 - 8 a$

Combine $2$ and $\frac{1}{3}$ .

$4 = \frac{2}{3} + 3$

$a = \frac{1}{12}$

$b = 3 - 8 a$

$4 = \frac{2}{3} + 3$

$a = \frac{1}{12}$

$b = 3 - 8 a$

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

$4 = \frac{2}{3} + 3 \cdot \frac{3}{3}$

$a = \frac{1}{12}$

$b = 3 - 8 a$

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

$4 = \frac{2}{3} + \frac{3 \cdot 3}{3}$

$a = \frac{1}{12}$

$b = 3 - 8 a$

Combine the numerators over the common denominator.

$4 = \frac{2 + 3 \cdot 3}{3}$

$a = \frac{1}{12}$

$b = 3 - 8 a$

Simplify the numerator.

Multiply $3$ by $3$ .

$4 = \frac{2 + 9}{3}$

$a = \frac{1}{12}$

$b = 3 - 8 a$

Add $2$ and $9$ .

$4 = \frac{11}{3}$

$a = \frac{1}{12}$

$b = 3 - 8 a$

$4 = \frac{11}{3}$

$a = \frac{1}{12}$

$b = 3 - 8 a$

$4 = \frac{11}{3}$

$a = \frac{1}{12}$

$b = 3 - 8 a$

$4 = \frac{11}{3}$

$a = \frac{1}{12}$

$b = 3 - 8 a$

Replace all occurrences of $a$ in $b = 3 - 8 a$ with $\frac{1}{12}$ .

$b = 3 - 8 (\frac{1}{12})$

$4 = \frac{11}{3}$

$a = \frac{1}{12}$

Simplify the right side.

Simplify $3 - 8 (\frac{1}{12})$ .

Simplify each term.

Cancel the common factor of $4$ .

Factor $4$ out of $- 8$ .

$b = 3 + 4 (- 2) (\frac{1}{12})$

$4 = \frac{11}{3}$

$a = \frac{1}{12}$

Factor $4$ out of $12$ .

$b = 3 + 4 \cdot (- 2 \frac{1}{4 \cdot 3})$

$4 = \frac{11}{3}$

$a = \frac{1}{12}$

Cancel the common factor.

$b = 3 + 4 \cdot (- 2 \frac{1}{4 \cdot 3})$

$4 = \frac{11}{3}$

$a = \frac{1}{12}$

Rewrite the expression.

$b = 3 - 2 (\frac{1}{3})$

$4 = \frac{11}{3}$

$a = \frac{1}{12}$

$b = 3 - 2 (\frac{1}{3})$

$4 = \frac{11}{3}$

$a = \frac{1}{12}$

Combine $- 2$ and $\frac{1}{3}$ .

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

$4 = \frac{11}{3}$

$a = \frac{1}{12}$

Move the negative in front of the fraction.

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

$4 = \frac{11}{3}$

$a = \frac{1}{12}$

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

$4 = \frac{11}{3}$

$a = \frac{1}{12}$

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

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

$4 = \frac{11}{3}$

$a = \frac{1}{12}$

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

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

$4 = \frac{11}{3}$

$a = \frac{1}{12}$

Combine the numerators over the common denominator.

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

$4 = \frac{11}{3}$

$a = \frac{1}{12}$

Simplify the numerator.

Multiply $3$ by $3$ .

$b = \frac{9 - 2}{3}$

$4 = \frac{11}{3}$

$a = \frac{1}{12}$

Subtract $2$ from $9$ .

$b = \frac{7}{3}$

$4 = \frac{11}{3}$

$a = \frac{1}{12}$

$b = \frac{7}{3}$

$4 = \frac{11}{3}$

$a = \frac{1}{12}$

$b = \frac{7}{3}$

$4 = \frac{11}{3}$

$a = \frac{1}{12}$

$b = \frac{7}{3}$

$4 = \frac{11}{3}$

$a = \frac{1}{12}$

$b = \frac{7}{3}$

$4 = \frac{11}{3}$

$a = \frac{1}{12}$

Since $4 = \frac{11}{3}$ is not true, there is no solution.

No solution

Since $y \neq f (x)$ 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 $f (x) = a x^{2} + b x + c$ .

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

Solve for $c$ in $3 = a \cdot 8^{2} + b (8) + c$ .

Rewrite the equation as $a \cdot 8^{2} + b (8) + c = 3$ .

$a \cdot 8^{2} + b (8) + c = 3$

$4 = a \cdot 16^{2} + b (16) + c$

$5 = a \cdot 32^{2} + b (32) + c$

Simplify each term.

Raise $8$ to the power of $2$ .

$a \cdot 64 + b (8) + c = 3$

$4 = a \cdot 16^{2} + b (16) + c$

$5 = a \cdot 32^{2} + b (32) + c$

Move $64$ to the left of $a$ .

$64 \cdot a + b (8) + c = 3$

$4 = a \cdot 16^{2} + b (16) + c$

$5 = a \cdot 32^{2} + b (32) + c$

Move $8$ to the left of $b$ .

$64 a + 8 b + c = 3$

$4 = a \cdot 16^{2} + b (16) + c$

$5 = a \cdot 32^{2} + b (32) + c$

$64 a + 8 b + c = 3$

$4 = a \cdot 16^{2} + b (16) + c$

$5 = a \cdot 32^{2} + b (32) + c$

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

Subtract $64 a$ from both sides of the equation.

$8 b + c = 3 - 64 a$

$4 = a \cdot 16^{2} + b (16) + c$

$5 = a \cdot 32^{2} + b (32) + c$

Subtract $8 b$ from both sides of the equation.

$c = 3 - 64 a - 8 b$

$4 = a \cdot 16^{2} + b (16) + c$

$5 = a \cdot 32^{2} + b (32) + c$

$c = 3 - 64 a - 8 b$

$4 = a \cdot 16^{2} + b (16) + c$

$5 = a \cdot 32^{2} + b (32) + c$

$c = 3 - 64 a - 8 b$

$4 = a \cdot 16^{2} + b (16) + c$

$5 = a \cdot 32^{2} + b (32) + c$

Replace all occurrences of $c$ with $3 - 64 a - 8 b$ in each equation.

Replace all occurrences of $c$ in $4 = a \cdot 16^{2} + b (16) + c$ with $3 - 64 a - 8 b$ .

$4 = a \cdot 16^{2} + b (16) + 3 - 64 a - 8 b$

$c = 3 - 64 a - 8 b$

$5 = a \cdot 32^{2} + b (32) + c$

Simplify $4 = a \cdot 16^{2} + b (16) + 3 - 64 a - 8 b$ .

Simplify the left side.

Remove parentheses.

$4 = a \cdot 16^{2} + b (16) + 3 - 64 a - 8 b$

$c = 3 - 64 a - 8 b$

$5 = a \cdot 32^{2} + b (32) + c$

$4 = a \cdot 16^{2} + b (16) + 3 - 64 a - 8 b$

$c = 3 - 64 a - 8 b$

$5 = a \cdot 32^{2} + b (32) + c$

Simplify the right side.

Simplify $a \cdot 16^{2} + b (16) + 3 - 64 a - 8 b$ .

Simplify each term.

Raise $16$ to the power of $2$ .

$4 = a \cdot 256 + b (16) + 3 - 64 a - 8 b$

$c = 3 - 64 a - 8 b$

$5 = a \cdot 32^{2} + b (32) + c$

Move $256$ to the left of $a$ .

$4 = 256 \cdot a + b (16) + 3 - 64 a - 8 b$

$c = 3 - 64 a - 8 b$

$5 = a \cdot 32^{2} + b (32) + c$

Move $16$ to the left of $b$ .

$4 = 256 a + 16 b + 3 - 64 a - 8 b$

$c = 3 - 64 a - 8 b$

$5 = a \cdot 32^{2} + b (32) + c$

$4 = 256 a + 16 b + 3 - 64 a - 8 b$

$c = 3 - 64 a - 8 b$

$5 = a \cdot 32^{2} + b (32) + c$

Simplify by adding terms.

Subtract $64 a$ from $256 a$ .

$4 = 192 a + 16 b + 3 - 8 b$

$c = 3 - 64 a - 8 b$

$5 = a \cdot 32^{2} + b (32) + c$

Subtract $8 b$ from $16 b$ .

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

$5 = a \cdot 32^{2} + b (32) + c$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

$5 = a \cdot 32^{2} + b (32) + c$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

$5 = a \cdot 32^{2} + b (32) + c$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

$5 = a \cdot 32^{2} + b (32) + c$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

$5 = a \cdot 32^{2} + b (32) + c$

Replace all occurrences of $c$ in $5 = a \cdot 32^{2} + b (32) + c$ with $3 - 64 a - 8 b$ .

$5 = a \cdot 32^{2} + b (32) + 3 - 64 a - 8 b$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

Simplify $5 = a \cdot 32^{2} + b (32) + 3 - 64 a - 8 b$ .

Simplify the left side.

Remove parentheses.

$5 = a \cdot 32^{2} + b (32) + 3 - 64 a - 8 b$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

$5 = a \cdot 32^{2} + b (32) + 3 - 64 a - 8 b$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

Simplify the right side.

Simplify $a \cdot 32^{2} + b (32) + 3 - 64 a - 8 b$ .

Simplify each term.

Raise $32$ to the power of $2$ .

$5 = a \cdot 1024 + b (32) + 3 - 64 a - 8 b$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

Move $1024$ to the left of $a$ .

$5 = 1024 \cdot a + b (32) + 3 - 64 a - 8 b$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

Move $32$ to the left of $b$ .

$5 = 1024 a + 32 b + 3 - 64 a - 8 b$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

$5 = 1024 a + 32 b + 3 - 64 a - 8 b$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

Simplify by adding terms.

Subtract $64 a$ from $1024 a$ .

$5 = 960 a + 32 b + 3 - 8 b$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

Subtract $8 b$ from $32 b$ .

$5 = 960 a + 24 b + 3$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

$5 = 960 a + 24 b + 3$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

$5 = 960 a + 24 b + 3$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

$5 = 960 a + 24 b + 3$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

$5 = 960 a + 24 b + 3$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

$5 = 960 a + 24 b + 3$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

Solve for $a$ in $5 = 960 a + 24 b + 3$ .

Rewrite the equation as $960 a + 24 b + 3 = 5$ .

$960 a + 24 b + 3 = 5$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

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

Subtract $24 b$ from both sides of the equation.

$960 a + 3 = 5 - 24 b$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

Subtract $3$ from both sides of the equation.

$960 a = 5 - 24 b - 3$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

Subtract $3$ from $5$ .

$960 a = - 24 b + 2$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

$960 a = - 24 b + 2$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

Divide each term in $960 a = - 24 b + 2$ by $960$ and simplify.

Divide each term in $960 a = - 24 b + 2$ by $960$ .

$\frac{960 a}{960} = \frac{- 24 b}{960} + \frac{2}{960}$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

Simplify the left side.

Cancel the common factor of $960$ .

Cancel the common factor.

$\frac{960 a}{960} = \frac{- 24 b}{960} + \frac{2}{960}$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

Divide $a$ by $1$ .

$a = \frac{- 24 b}{960} + \frac{2}{960}$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

$a = \frac{- 24 b}{960} + \frac{2}{960}$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

$a = \frac{- 24 b}{960} + \frac{2}{960}$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 24$ and $960$ .

Factor $24$ out of $- 24 b$ .

$a = \frac{24 (- b)}{960} + \frac{2}{960}$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

Cancel the common factors.

Factor $24$ out of $960$ .

$a = \frac{24 (- b)}{24 (40)} + \frac{2}{960}$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

Cancel the common factor.

$a = \frac{24 (- b)}{24 \cdot 40} + \frac{2}{960}$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

Rewrite the expression.

$a = \frac{- b}{40} + \frac{2}{960}$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

$a = \frac{- b}{40} + \frac{2}{960}$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

$a = \frac{- b}{40} + \frac{2}{960}$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

Move the negative in front of the fraction.

$a = - \frac{b}{40} + \frac{2}{960}$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

Cancel the common factor of $2$ and $960$ .

Factor $2$ out of $2$ .

$a = - \frac{b}{40} + \frac{2 (1)}{960}$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

Cancel the common factors.

Factor $2$ out of $960$ .

$a = - \frac{b}{40} + \frac{2 \cdot 1}{2 \cdot 480}$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

Cancel the common factor.

$a = - \frac{b}{40} + \frac{2 \cdot 1}{2 \cdot 480}$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

Rewrite the expression.

$a = - \frac{b}{40} + \frac{1}{480}$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

$a = - \frac{b}{40} + \frac{1}{480}$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

$a = - \frac{b}{40} + \frac{1}{480}$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

$a = - \frac{b}{40} + \frac{1}{480}$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

$a = - \frac{b}{40} + \frac{1}{480}$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

$a = - \frac{b}{40} + \frac{1}{480}$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

$a = - \frac{b}{40} + \frac{1}{480}$

$4 = 192 a + 8 b + 3$

$c = 3 - 64 a - 8 b$

Replace all occurrences of $a$ with $- \frac{b}{40} + \frac{1}{480}$ in each equation.

Replace all occurrences of $a$ in $4 = 192 a + 8 b + 3$ with $- \frac{b}{40} + \frac{1}{480}$ .

$4 = 192 (- \frac{b}{40} + \frac{1}{480}) + 8 b + 3$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

Simplify the right side.

Simplify $192 (- \frac{b}{40} + \frac{1}{480}) + 8 b + 3$ .

Simplify each term.

Apply the distributive property.

$4 = 192 (- \frac{b}{40}) + 192 (\frac{1}{480}) + 8 b + 3$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

Cancel the common factor of $8$ .

Move the leading negative in $- \frac{b}{40}$ into the numerator.

$4 = 192 (\frac{- b}{40}) + 192 (\frac{1}{480}) + 8 b + 3$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

Factor $8$ out of $192$ .

$4 = 8 (24) (\frac{- b}{40}) + 192 (\frac{1}{480}) + 8 b + 3$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

Factor $8$ out of $40$ .

$4 = 8 \cdot (24 (\frac{- b}{8 \cdot 5})) + 192 (\frac{1}{480}) + 8 b + 3$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

Cancel the common factor.

$4 = 8 \cdot (24 (\frac{- b}{8 \cdot 5})) + 192 (\frac{1}{480}) + 8 b + 3$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

Rewrite the expression.

$4 = 24 (\frac{- b}{5}) + 192 (\frac{1}{480}) + 8 b + 3$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

$4 = 24 (\frac{- b}{5}) + 192 (\frac{1}{480}) + 8 b + 3$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

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

$4 = \frac{24 (- b)}{5} + 192 (\frac{1}{480}) + 8 b + 3$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

Multiply $- 1$ by $24$ .

$4 = \frac{- 24 b}{5} + 192 (\frac{1}{480}) + 8 b + 3$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

Cancel the common factor of $96$ .

Factor $96$ out of $192$ .

$4 = \frac{- 24 b}{5} + 96 (2) (\frac{1}{480}) + 8 b + 3$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

Factor $96$ out of $480$ .

$4 = \frac{- 24 b}{5} + 96 \cdot (2 (\frac{1}{96 \cdot 5})) + 8 b + 3$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

Cancel the common factor.

$4 = \frac{- 24 b}{5} + 96 \cdot (2 (\frac{1}{96 \cdot 5})) + 8 b + 3$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

Rewrite the expression.

$4 = \frac{- 24 b}{5} + 2 (\frac{1}{5}) + 8 b + 3$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

$4 = \frac{- 24 b}{5} + 2 (\frac{1}{5}) + 8 b + 3$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

Combine $2$ and $\frac{1}{5}$ .

$4 = \frac{- 24 b}{5} + \frac{2}{5} + 8 b + 3$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

Move the negative in front of the fraction.

$4 = - \frac{24 b}{5} + \frac{2}{5} + 8 b + 3$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

$4 = - \frac{24 b}{5} + \frac{2}{5} + 8 b + 3$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

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

$4 = - \frac{24 b}{5} + 8 b \cdot \frac{5}{5} + \frac{2}{5} + 3$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

Combine $8 b$ and $\frac{5}{5}$ .

$4 = - \frac{24 b}{5} + \frac{8 b \cdot 5}{5} + \frac{2}{5} + 3$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

Combine the numerators over the common denominator.

$4 = \frac{- 24 b + 8 b \cdot 5}{5} + \frac{2}{5} + 3$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

Combine the numerators over the common denominator.

$4 = 3 + \frac{- 24 b + 8 b \cdot 5 + 2}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

Multiply $5$ by $8$ .

$4 = 3 + \frac{- 24 b + 40 b + 2}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

Add $- 24 b$ and $40 b$ .

$4 = 3 + \frac{16 b + 2}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

Factor $2$ out of $16 b + 2$ .

Factor $2$ out of $16 b$ .

$4 = 3 + \frac{2 (8 b) + 2}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

Factor $2$ out of $2$ .

$4 = 3 + \frac{2 (8 b) + 2 (1)}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

Factor $2$ out of $2 (8 b) + 2 (1)$ .

$4 = 3 + \frac{2 (8 b + 1)}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

$4 = 3 + \frac{2 (8 b + 1)}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

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

$4 = 3 \cdot \frac{5}{5} + \frac{2 (8 b + 1)}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

Simplify terms.

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

$4 = \frac{3 \cdot 5}{5} + \frac{2 (8 b + 1)}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

Combine the numerators over the common denominator.

$4 = \frac{3 \cdot 5 + 2 (8 b + 1)}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

$4 = \frac{3 \cdot 5 + 2 (8 b + 1)}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

Simplify the numerator.

Multiply $3$ by $5$ .

$4 = \frac{15 + 2 (8 b + 1)}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

Apply the distributive property.

$4 = \frac{15 + 2 (8 b) + 2 \cdot 1}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

Multiply $8$ by $2$ .

$4 = \frac{15 + 16 b + 2 \cdot 1}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

Multiply $2$ by $1$ .

$4 = \frac{15 + 16 b + 2}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

Add $15$ and $2$ .

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 64 a - 8 b$

Replace all occurrences of $a$ in $c = 3 - 64 a - 8 b$ with $- \frac{b}{40} + \frac{1}{480}$ .

$c = 3 - 64 (- \frac{b}{40} + \frac{1}{480}) - 8 b$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

Simplify the right side.

Simplify $3 - 64 (- \frac{b}{40} + \frac{1}{480}) - 8 b$ .

Simplify each term.

Apply the distributive property.

$c = 3 - 64 (- \frac{b}{40}) - 64 (\frac{1}{480}) - 8 b$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

Cancel the common factor of $8$ .

Move the leading negative in $- \frac{b}{40}$ into the numerator.

$c = 3 - 64 \frac{- b}{40} - 64 (\frac{1}{480}) - 8 b$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

Factor $8$ out of $- 64$ .

$c = 3 + 8 (- 8) (\frac{- b}{40}) - 64 (\frac{1}{480}) - 8 b$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

Factor $8$ out of $40$ .

$c = 3 + 8 \cdot (- 8 \frac{- b}{8 \cdot 5}) - 64 (\frac{1}{480}) - 8 b$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

Cancel the common factor.

$c = 3 + 8 \cdot (- 8 \frac{- b}{8 \cdot 5}) - 64 (\frac{1}{480}) - 8 b$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

Rewrite the expression.

$c = 3 - 8 \frac{- b}{5} - 64 (\frac{1}{480}) - 8 b$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 - 8 \frac{- b}{5} - 64 (\frac{1}{480}) - 8 b$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

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

$c = 3 + \frac{- 8 (- b)}{5} - 64 (\frac{1}{480}) - 8 b$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

Multiply $- 1$ by $- 8$ .

$c = 3 + \frac{8 b}{5} - 64 (\frac{1}{480}) - 8 b$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

Cancel the common factor of $32$ .

Factor $32$ out of $- 64$ .

$c = 3 + \frac{8 b}{5} + 32 (- 2) (\frac{1}{480}) - 8 b$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

Factor $32$ out of $480$ .

$c = 3 + \frac{8 b}{5} + 32 \cdot (- 2 \frac{1}{32 \cdot 15}) - 8 b$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

Cancel the common factor.

$c = 3 + \frac{8 b}{5} + 32 \cdot (- 2 \frac{1}{32 \cdot 15}) - 8 b$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

Rewrite the expression.

$c = 3 + \frac{8 b}{5} - 2 (\frac{1}{15}) - 8 b$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 + \frac{8 b}{5} - 2 (\frac{1}{15}) - 8 b$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

Combine $- 2$ and $\frac{1}{15}$ .

$c = 3 + \frac{8 b}{5} + \frac{- 2}{15} - 8 b$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

Move the negative in front of the fraction.

$c = 3 + \frac{8 b}{5} - \frac{2}{15} - 8 b$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = 3 + \frac{8 b}{5} - \frac{2}{15} - 8 b$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

To write $3$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$c = \frac{8 b}{5} + 3 \cdot \frac{15}{15} - \frac{2}{15} - 8 b$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

Combine $3$ and $\frac{15}{15}$ .

$c = \frac{8 b}{5} + \frac{3 \cdot 15}{15} - \frac{2}{15} - 8 b$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

Combine the numerators over the common denominator.

$c = \frac{8 b}{5} + \frac{3 \cdot 15 - 2}{15} - 8 b$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

Simplify the numerator.

Multiply $3$ by $15$ .

$c = \frac{8 b}{5} + \frac{45 - 2}{15} - 8 b$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

Subtract $2$ from $45$ .

$c = \frac{8 b}{5} + \frac{43}{15} - 8 b$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = \frac{8 b}{5} + \frac{43}{15} - 8 b$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

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

$c = \frac{8 b}{5} - 8 b \cdot \frac{5}{5} + \frac{43}{15}$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

Simplify terms.

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

$c = \frac{8 b}{5} + \frac{- 8 b \cdot 5}{5} + \frac{43}{15}$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

Combine the numerators over the common denominator.

$c = \frac{8 b - 8 b \cdot 5}{5} + \frac{43}{15}$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = \frac{8 b - 8 b \cdot 5}{5} + \frac{43}{15}$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

Simplify each term.

Simplify the numerator.

Factor $8 b$ out of $8 b - 8 b \cdot 5$ .

Factor $8 b$ out of $8 b$ .

$c = \frac{8 b (1) - 8 b \cdot 5}{5} + \frac{43}{15}$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

Factor $8 b$ out of $- 8 b \cdot 5$ .

$c = \frac{8 b (1) + 8 b (- 1 \cdot 5)}{5} + \frac{43}{15}$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

Factor $8 b$ out of $8 b (1) + 8 b (- 1 \cdot 5)$ .

$c = \frac{8 b (1 - 1 \cdot 5)}{5} + \frac{43}{15}$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = \frac{8 b (1 - 1 \cdot 5)}{5} + \frac{43}{15}$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

Multiply $- 1$ by $5$ .

$c = \frac{8 b (1 - 5)}{5} + \frac{43}{15}$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

Subtract $5$ from $1$ .

$c = \frac{8 b \cdot - 4}{5} + \frac{43}{15}$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

Multiply $- 4$ by $8$ .

$c = \frac{- 32 b}{5} + \frac{43}{15}$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = \frac{- 32 b}{5} + \frac{43}{15}$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

Move the negative in front of the fraction.

$c = - \frac{32 b}{5} + \frac{43}{15}$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = - \frac{32 b}{5} + \frac{43}{15}$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = - \frac{32 b}{5} + \frac{43}{15}$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = - \frac{32 b}{5} + \frac{43}{15}$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = - \frac{32 b}{5} + \frac{43}{15}$

$4 = \frac{16 b + 17}{5}$

$a = - \frac{b}{40} + \frac{1}{480}$

Solve for $b$ in $4 = \frac{16 b + 17}{5}$ .

Rewrite the equation as $\frac{16 b + 17}{5} = 4$ .

$\frac{16 b + 17}{5} = 4$

$c = - \frac{32 b}{5} + \frac{43}{15}$

$a = - \frac{b}{40} + \frac{1}{480}$

Multiply both sides by $5$ .

$\frac{16 b + 17}{5} \cdot 5 = 4 \cdot 5$

$c = - \frac{32 b}{5} + \frac{43}{15}$

$a = - \frac{b}{40} + \frac{1}{480}$

Simplify.

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{16 b + 17}{5} \cdot 5 = 4 \cdot 5$

$c = - \frac{32 b}{5} + \frac{43}{15}$

$a = - \frac{b}{40} + \frac{1}{480}$

Rewrite the expression.

$16 b + 17 = 4 \cdot 5$

$c = - \frac{32 b}{5} + \frac{43}{15}$

$a = - \frac{b}{40} + \frac{1}{480}$

$16 b + 17 = 4 \cdot 5$

$c = - \frac{32 b}{5} + \frac{43}{15}$

$a = - \frac{b}{40} + \frac{1}{480}$

$16 b + 17 = 4 \cdot 5$

$c = - \frac{32 b}{5} + \frac{43}{15}$

$a = - \frac{b}{40} + \frac{1}{480}$

Simplify the right side.

Multiply $4$ by $5$ .

$16 b + 17 = 20$

$c = - \frac{32 b}{5} + \frac{43}{15}$

$a = - \frac{b}{40} + \frac{1}{480}$

$16 b + 17 = 20$

$c = - \frac{32 b}{5} + \frac{43}{15}$

$a = - \frac{b}{40} + \frac{1}{480}$

$16 b + 17 = 20$

$c = - \frac{32 b}{5} + \frac{43}{15}$

$a = - \frac{b}{40} + \frac{1}{480}$

Solve for $b$ .

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

Subtract $17$ from both sides of the equation.

$16 b = 20 - 17$

$c = - \frac{32 b}{5} + \frac{43}{15}$

$a = - \frac{b}{40} + \frac{1}{480}$

Subtract $17$ from $20$ .

$16 b = 3$

$c = - \frac{32 b}{5} + \frac{43}{15}$

$a = - \frac{b}{40} + \frac{1}{480}$

$16 b = 3$

$c = - \frac{32 b}{5} + \frac{43}{15}$

$a = - \frac{b}{40} + \frac{1}{480}$

Divide each term in $16 b = 3$ by $16$ and simplify.

Divide each term in $16 b = 3$ by $16$ .

$\frac{16 b}{16} = \frac{3}{16}$

$c = - \frac{32 b}{5} + \frac{43}{15}$

$a = - \frac{b}{40} + \frac{1}{480}$

Simplify the left side.

Cancel the common factor of $16$ .

Cancel the common factor.

$\frac{16 b}{16} = \frac{3}{16}$

$c = - \frac{32 b}{5} + \frac{43}{15}$

$a = - \frac{b}{40} + \frac{1}{480}$

Divide $b$ by $1$ .

$b = \frac{3}{16}$

$c = - \frac{32 b}{5} + \frac{43}{15}$

$a = - \frac{b}{40} + \frac{1}{480}$

$b = \frac{3}{16}$

$c = - \frac{32 b}{5} + \frac{43}{15}$

$a = - \frac{b}{40} + \frac{1}{480}$

$b = \frac{3}{16}$

$c = - \frac{32 b}{5} + \frac{43}{15}$

$a = - \frac{b}{40} + \frac{1}{480}$

$b = \frac{3}{16}$

$c = - \frac{32 b}{5} + \frac{43}{15}$

$a = - \frac{b}{40} + \frac{1}{480}$

$b = \frac{3}{16}$

$c = - \frac{32 b}{5} + \frac{43}{15}$

$a = - \frac{b}{40} + \frac{1}{480}$

$b = \frac{3}{16}$

$c = - \frac{32 b}{5} + \frac{43}{15}$

$a = - \frac{b}{40} + \frac{1}{480}$

Replace all occurrences of $b$ with $\frac{3}{16}$ in each equation.

Replace all occurrences of $b$ in $c = - \frac{32 b}{5} + \frac{43}{15}$ with $\frac{3}{16}$ .

$c = - \frac{32 (\frac{3}{16})}{5} + \frac{43}{15}$

$b = \frac{3}{16}$

$a = - \frac{b}{40} + \frac{1}{480}$

Simplify the right side.

Simplify $- \frac{32 (\frac{3}{16})}{5} + \frac{43}{15}$ .

Simplify each term.

Combine $32$ and $\frac{3}{16}$ .

$c = - \frac{\frac{32 \cdot 3}{16}}{5} + \frac{43}{15}$

$b = \frac{3}{16}$

$a = - \frac{b}{40} + \frac{1}{480}$

Multiply $32$ by $3$ .

$c = - \frac{\frac{96}{16}}{5} + \frac{43}{15}$

$b = \frac{3}{16}$

$a = - \frac{b}{40} + \frac{1}{480}$

Divide $96$ by $16$ .

$c = - \frac{6}{5} + \frac{43}{15}$

$b = \frac{3}{16}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = - \frac{6}{5} + \frac{43}{15}$

$b = \frac{3}{16}$

$a = - \frac{b}{40} + \frac{1}{480}$

To write $- \frac{6}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$c = - \frac{6}{5} \cdot \frac{3}{3} + \frac{43}{15}$

$b = \frac{3}{16}$

$a = - \frac{b}{40} + \frac{1}{480}$

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{6}{5}$ by $\frac{3}{3}$ .

$c = - \frac{6 \cdot 3}{5 \cdot 3} + \frac{43}{15}$

$b = \frac{3}{16}$

$a = - \frac{b}{40} + \frac{1}{480}$

Multiply $5$ by $3$ .

$c = - \frac{6 \cdot 3}{15} + \frac{43}{15}$

$b = \frac{3}{16}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = - \frac{6 \cdot 3}{15} + \frac{43}{15}$

$b = \frac{3}{16}$

$a = - \frac{b}{40} + \frac{1}{480}$

Combine the numerators over the common denominator.

$c = \frac{- 6 \cdot 3 + 43}{15}$

$b = \frac{3}{16}$

$a = - \frac{b}{40} + \frac{1}{480}$

Simplify the numerator.

Multiply $- 6$ by $3$ .

$c = \frac{- 18 + 43}{15}$

$b = \frac{3}{16}$

$a = - \frac{b}{40} + \frac{1}{480}$

Add $- 18$ and $43$ .

$c = \frac{25}{15}$

$b = \frac{3}{16}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = \frac{25}{15}$

$b = \frac{3}{16}$

$a = - \frac{b}{40} + \frac{1}{480}$

Cancel the common factor of $25$ and $15$ .

Factor $5$ out of $25$ .

$c = \frac{5 (5)}{15}$

$b = \frac{3}{16}$

$a = - \frac{b}{40} + \frac{1}{480}$

Cancel the common factors.

Factor $5$ out of $15$ .

$c = \frac{5 \cdot 5}{5 \cdot 3}$

$b = \frac{3}{16}$

$a = - \frac{b}{40} + \frac{1}{480}$

Cancel the common factor.

$c = \frac{5 \cdot 5}{5 \cdot 3}$

$b = \frac{3}{16}$

$a = - \frac{b}{40} + \frac{1}{480}$

Rewrite the expression.

$c = \frac{5}{3}$

$b = \frac{3}{16}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = \frac{5}{3}$

$b = \frac{3}{16}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = \frac{5}{3}$

$b = \frac{3}{16}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = \frac{5}{3}$

$b = \frac{3}{16}$

$a = - \frac{b}{40} + \frac{1}{480}$

$c = \frac{5}{3}$

$b = \frac{3}{16}$

$a = - \frac{b}{40} + \frac{1}{480}$

Replace all occurrences of $b$ in $a = - \frac{b}{40} + \frac{1}{480}$ with $\frac{3}{16}$ .

$a = - \frac{\frac{3}{16}}{40} + \frac{1}{480}$

$c = \frac{5}{3}$

$b = \frac{3}{16}$

Simplify the right side.

Simplify $- \frac{\frac{3}{16}}{40} + \frac{1}{480}$ .

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

$a = - (\frac{3}{16} \cdot \frac{1}{40}) + \frac{1}{480}$

$c = \frac{5}{3}$

$b = \frac{3}{16}$

Multiply $\frac{3}{16} \cdot \frac{1}{40}$ .

Multiply $\frac{3}{16}$ by $\frac{1}{40}$ .

$a = - \frac{3}{16 \cdot 40} + \frac{1}{480}$

$c = \frac{5}{3}$

$b = \frac{3}{16}$

Multiply $16$ by $40$ .

$a = - \frac{3}{640} + \frac{1}{480}$

$c = \frac{5}{3}$

$b = \frac{3}{16}$

$a = - \frac{3}{640} + \frac{1}{480}$

$c = \frac{5}{3}$

$b = \frac{3}{16}$

$a = - \frac{3}{640} + \frac{1}{480}$

$c = \frac{5}{3}$

$b = \frac{3}{16}$

To write $- \frac{3}{640}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$a = - \frac{3}{640} \cdot \frac{3}{3} + \frac{1}{480}$

$c = \frac{5}{3}$

$b = \frac{3}{16}$

To write $\frac{1}{480}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$a = - \frac{3}{640} \cdot \frac{3}{3} + \frac{1}{480} \cdot \frac{4}{4}$

$c = \frac{5}{3}$

$b = \frac{3}{16}$

Write each expression with a common denominator of $1920$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3}{640}$ by $\frac{3}{3}$ .

$a = - \frac{3 \cdot 3}{640 \cdot 3} + \frac{1}{480} \cdot \frac{4}{4}$

$c = \frac{5}{3}$

$b = \frac{3}{16}$

Multiply $640$ by $3$ .

$a = - \frac{3 \cdot 3}{1920} + \frac{1}{480} \cdot \frac{4}{4}$

$c = \frac{5}{3}$

$b = \frac{3}{16}$

Multiply $\frac{1}{480}$ by $\frac{4}{4}$ .

$a = - \frac{3 \cdot 3}{1920} + \frac{4}{480 \cdot 4}$

$c = \frac{5}{3}$

$b = \frac{3}{16}$

Multiply $480$ by $4$ .

$a = - \frac{3 \cdot 3}{1920} + \frac{4}{1920}$

$c = \frac{5}{3}$

$b = \frac{3}{16}$

$a = - \frac{3 \cdot 3}{1920} + \frac{4}{1920}$

$c = \frac{5}{3}$

$b = \frac{3}{16}$

Combine the numerators over the common denominator.

$a = \frac{- 3 \cdot 3 + 4}{1920}$

$c = \frac{5}{3}$

$b = \frac{3}{16}$

Simplify the numerator.

Multiply $- 3$ by $3$ .

$a = \frac{- 9 + 4}{1920}$

$c = \frac{5}{3}$

$b = \frac{3}{16}$

Add $- 9$ and $4$ .

$a = \frac{- 5}{1920}$

$c = \frac{5}{3}$

$b = \frac{3}{16}$

$a = \frac{- 5}{1920}$

$c = \frac{5}{3}$

$b = \frac{3}{16}$

Cancel the common factor of $- 5$ and $1920$ .

Factor $5$ out of $- 5$ .

$a = \frac{5 (- 1)}{1920}$

$c = \frac{5}{3}$

$b = \frac{3}{16}$

Cancel the common factors.

Factor $5$ out of $1920$ .

$a = \frac{5 \cdot - 1}{5 \cdot 384}$

$c = \frac{5}{3}$

$b = \frac{3}{16}$

Cancel the common factor.

$a = \frac{5 \cdot - 1}{5 \cdot 384}$

$c = \frac{5}{3}$

$b = \frac{3}{16}$

Rewrite the expression.

$a = \frac{- 1}{384}$

$c = \frac{5}{3}$

$b = \frac{3}{16}$

$a = \frac{- 1}{384}$

$c = \frac{5}{3}$

$b = \frac{3}{16}$

$a = \frac{- 1}{384}$

$c = \frac{5}{3}$

$b = \frac{3}{16}$

Move the negative in front of the fraction.

$a = - \frac{1}{384}$

$c = \frac{5}{3}$

$b = \frac{3}{16}$

$a = - \frac{1}{384}$

$c = \frac{5}{3}$

$b = \frac{3}{16}$

$a = - \frac{1}{384}$

$c = \frac{5}{3}$

$b = \frac{3}{16}$

$a = - \frac{1}{384}$

$c = \frac{5}{3}$

$b = \frac{3}{16}$

List all of the solutions.

$a = - \frac{1}{384}, c = \frac{5}{3}, b = \frac{3}{16}$

Calculate the value of $y$ using each $x$ value in the table and compare this value to the given $f (x)$ value in the table.

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = - \frac{1}{384}$ , $b = \frac{3}{16}$ , $c = \frac{5}{3}$ , and $x = 8$ .

Simplify each term.

Raise $8$ to the power of $2$ .

$y = - \frac{1}{384} \cdot 64 + (\frac{3}{16}) \cdot (8) + \frac{5}{3}$

Cancel the common factor of $64$ .

Move the leading negative in $- \frac{1}{384}$ into the numerator.

$y = \frac{- 1}{384} \cdot 64 + (\frac{3}{16}) \cdot (8) + \frac{5}{3}$

Factor $64$ out of $384$ .

$y = \frac{- 1}{64 (6)} \cdot 64 + (\frac{3}{16}) \cdot (8) + \frac{5}{3}$

Cancel the common factor.

$y = \frac{- 1}{64 \cdot 6} \cdot 64 + (\frac{3}{16}) \cdot (8) + \frac{5}{3}$

Rewrite the expression.

$y = \frac{- 1}{6} + (\frac{3}{16}) \cdot (8) + \frac{5}{3}$

Move the negative in front of the fraction.

$y = - \frac{1}{6} + (\frac{3}{16}) \cdot (8) + \frac{5}{3}$

Cancel the common factor of $8$ .

Factor $8$ out of $16$ .

$y = - \frac{1}{6} + \frac{3}{8 (2)} \cdot 8 + \frac{5}{3}$

Cancel the common factor.

$y = - \frac{1}{6} + \frac{3}{8 \cdot 2} \cdot 8 + \frac{5}{3}$

Rewrite the expression.

$y = - \frac{1}{6} + \frac{3}{2} + \frac{5}{3}$

Find the common denominator.

Multiply $\frac{3}{2}$ by $\frac{3}{3}$ .

$y = - \frac{1}{6} + \frac{3}{2} \cdot \frac{3}{3} + \frac{5}{3}$

Multiply $\frac{3}{2}$ by $\frac{3}{3}$ .

$y = - \frac{1}{6} + \frac{3 \cdot 3}{2 \cdot 3} + \frac{5}{3}$

Multiply $\frac{5}{3}$ by $\frac{2}{2}$ .

$y = - \frac{1}{6} + \frac{3 \cdot 3}{2 \cdot 3} + \frac{5}{3} \cdot \frac{2}{2}$

Multiply $\frac{5}{3}$ by $\frac{2}{2}$ .

$y = - \frac{1}{6} + \frac{3 \cdot 3}{2 \cdot 3} + \frac{5 \cdot 2}{3 \cdot 2}$

Multiply $2$ by $3$ .

$y = - \frac{1}{6} + \frac{3 \cdot 3}{6} + \frac{5 \cdot 2}{3 \cdot 2}$

Reorder the factors of $3 \cdot 2$ .

$y = - \frac{1}{6} + \frac{3 \cdot 3}{6} + \frac{5 \cdot 2}{2 \cdot 3}$

Multiply $2$ by $3$ .

$y = - \frac{1}{6} + \frac{3 \cdot 3}{6} + \frac{5 \cdot 2}{6}$

Combine the numerators over the common denominator.

$y = \frac{- 1 + 3 \cdot 3 + 5 \cdot 2}{6}$

Simplify each term.

Multiply $3$ by $3$ .

$y = \frac{- 1 + 9 + 5 \cdot 2}{6}$

Multiply $5$ by $2$ .

$y = \frac{- 1 + 9 + 10}{6}$

Simplify the expression.

Add $- 1$ and $9$ .

$y = \frac{8 + 10}{6}$

Add $8$ and $10$ .

$y = \frac{18}{6}$

Divide $18$ by $6$ .

$y = 3$

If the table has a quadratic function rule, $y = f (x)$ for the corresponding $x$ value, $x = 8$ . This check passes since $y = 3$ and $f (x) = 3$ .

$3 = 3$

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = - \frac{1}{384}$ , $b = \frac{3}{16}$ , $c = \frac{5}{3}$ , and $x = 16$ .

Simplify each term.

Raise $16$ to the power of $2$ .

$y = - \frac{1}{384} \cdot 256 + (\frac{3}{16}) \cdot (16) + \frac{5}{3}$

Cancel the common factor of $128$ .

Move the leading negative in $- \frac{1}{384}$ into the numerator.

$y = \frac{- 1}{384} \cdot 256 + (\frac{3}{16}) \cdot (16) + \frac{5}{3}$

Factor $128$ out of $384$ .

$y = \frac{- 1}{128 (3)} \cdot 256 + (\frac{3}{16}) \cdot (16) + \frac{5}{3}$

Factor $128$ out of $256$ .

$y = \frac{- 1}{128 \cdot 3} \cdot (128 \cdot 2) + (\frac{3}{16}) \cdot (16) + \frac{5}{3}$

Cancel the common factor.

$y = \frac{- 1}{128 \cdot 3} \cdot (128 \cdot 2) + (\frac{3}{16}) \cdot (16) + \frac{5}{3}$

Rewrite the expression.

$y = \frac{- 1}{3} \cdot 2 + (\frac{3}{16}) \cdot (16) + \frac{5}{3}$

Combine $\frac{- 1}{3}$ and $2$ .

$y = \frac{- 1 \cdot 2}{3} + (\frac{3}{16}) \cdot (16) + \frac{5}{3}$

Multiply $- 1$ by $2$ .

$y = \frac{- 2}{3} + (\frac{3}{16}) \cdot (16) + \frac{5}{3}$

Move the negative in front of the fraction.

$y = - \frac{2}{3} + (\frac{3}{16}) \cdot (16) + \frac{5}{3}$

Cancel the common factor of $16$ .

Cancel the common factor.

$y = - \frac{2}{3} + \frac{3}{16} \cdot 16 + \frac{5}{3}$

Rewrite the expression.

$y = - \frac{2}{3} + 3 + \frac{5}{3}$

Combine fractions.

Combine the numerators over the common denominator.

$y = 3 + \frac{- 2 + 5}{3}$

Simplify the expression.

Add $- 2$ and $5$ .

$y = 3 + \frac{3}{3}$

Divide $3$ by $3$ .

$y = 3 + 1$

Add $3$ and $1$ .

$y = 4$

If the table has a quadratic function rule, $y = f (x)$ for the corresponding $x$ value, $x = 16$ . This check passes since $y = 4$ and $f (x) = 4$ .

$4 = 4$

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = - \frac{1}{384}$ , $b = \frac{3}{16}$ , $c = \frac{5}{3}$ , and $x = 32$ .

Simplify each term.

Raise $32$ to the power of $2$ .

$y = - \frac{1}{384} \cdot 1024 + (\frac{3}{16}) \cdot (32) + \frac{5}{3}$

Cancel the common factor of $128$ .

Move the leading negative in $- \frac{1}{384}$ into the numerator.

$y = \frac{- 1}{384} \cdot 1024 + (\frac{3}{16}) \cdot (32) + \frac{5}{3}$

Factor $128$ out of $384$ .

$y = \frac{- 1}{128 (3)} \cdot 1024 + (\frac{3}{16}) \cdot (32) + \frac{5}{3}$

Factor $128$ out of $1024$ .

$y = \frac{- 1}{128 \cdot 3} \cdot (128 \cdot 8) + (\frac{3}{16}) \cdot (32) + \frac{5}{3}$

Cancel the common factor.

$y = \frac{- 1}{128 \cdot 3} \cdot (128 \cdot 8) + (\frac{3}{16}) \cdot (32) + \frac{5}{3}$

Rewrite the expression.

$y = \frac{- 1}{3} \cdot 8 + (\frac{3}{16}) \cdot (32) + \frac{5}{3}$

Combine $\frac{- 1}{3}$ and $8$ .

$y = \frac{- 1 \cdot 8}{3} + (\frac{3}{16}) \cdot (32) + \frac{5}{3}$

Multiply $- 1$ by $8$ .

$y = \frac{- 8}{3} + (\frac{3}{16}) \cdot (32) + \frac{5}{3}$

Move the negative in front of the fraction.

$y = - \frac{8}{3} + (\frac{3}{16}) \cdot (32) + \frac{5}{3}$

Cancel the common factor of $16$ .

Factor $16$ out of $32$ .

$y = - \frac{8}{3} + \frac{3}{16} \cdot (16 (2)) + \frac{5}{3}$

Cancel the common factor.

$y = - \frac{8}{3} + \frac{3}{16} \cdot (16 \cdot 2) + \frac{5}{3}$

Rewrite the expression.

$y = - \frac{8}{3} + 3 \cdot 2 + \frac{5}{3}$

Multiply $3$ by $2$ .

$y = - \frac{8}{3} + 6 + \frac{5}{3}$

Combine fractions.

Combine the numerators over the common denominator.

$y = 6 + \frac{- 8 + 5}{3}$

Simplify the expression.

Add $- 8$ and $5$ .

$y = 6 + \frac{- 3}{3}$

Divide $- 3$ by $3$ .

$y = 6 - 1$

Subtract $1$ from $6$ .

$y = 5$

If the table has a quadratic function rule, $y = f (x)$ for the corresponding $x$ value, $x = 32$ . This check passes since $y = 5$ and $f (x) = 5$ .

$5 = 5$

Since $y = f (x)$ for the corresponding $x$ values, the function is quadratic.

The function is quadratic

Step 3

Since all $y = f (x)$ , the function is quadratic and follows the form $y = - \frac{x^{2}}{384} + \frac{3 x}{16} + \frac{5}{3}$ .

$y = - \frac{x^{2}}{384} + \frac{3 x}{16} + \frac{5}{3}$

Find the Function Rule table[[x,f(x)],[8,3],[16,4],[32,5]]

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