Find the Function Rule table[[x,y],[1,-10],[2,-20],[3,-40]]

$\begin{array}{c} x & y \\ 1 & - 10 \\ 2 & - 20 \\ 3 & - 40 \end{array}$

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 $y = a x + b$ .

$- 10 = a (1) + b - 20 = a (2) + b - 40 = a (3) + b$

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

Solve for $a$ in $- 10 = a + b$ .

Rewrite the equation as $a + b = - 10$ .

$a + b = - 10$

$- 20 = a (2) + b$

$- 40 = a (3) + b$

Subtract $b$ from both sides of the equation.

$a = - 10 - b$

$- 20 = a (2) + b$

$- 40 = a (3) + b$

$a = - 10 - b$

$- 20 = a (2) + b$

$- 40 = a (3) + b$

Replace all occurrences of $a$ with $- 10 - b$ in each equation.

Replace all occurrences of $a$ in $- 20 = a (2) + b$ with $- 10 - b$ .

$- 20 = (- 10 - b) (2) + b$

$a = - 10 - b$

$- 40 = a (3) + b$

Simplify the right side.

Simplify $(- 10 - b) (2) + b$ .

Simplify each term.

Apply the distributive property.

$- 20 = - 10 \cdot 2 - b \cdot 2 + b$

$a = - 10 - b$

$- 40 = a (3) + b$

Multiply $- 10$ by $2$ .

$- 20 = - 20 - b \cdot 2 + b$

$a = - 10 - b$

$- 40 = a (3) + b$

Multiply $2$ by $- 1$ .

$- 20 = - 20 - 2 b + b$

$a = - 10 - b$

$- 40 = a (3) + b$

$- 20 = - 20 - 2 b + b$

$a = - 10 - b$

$- 40 = a (3) + b$

Add $- 2 b$ and $b$ .

$- 20 = - 20 - b$

$a = - 10 - b$

$- 40 = a (3) + b$

$- 20 = - 20 - b$

$a = - 10 - b$

$- 40 = a (3) + b$

$- 20 = - 20 - b$

$a = - 10 - b$

$- 40 = a (3) + b$

Replace all occurrences of $a$ in $- 40 = a (3) + b$ with $- 10 - b$ .

$- 40 = (- 10 - b) (3) + b$

$- 20 = - 20 - b$

$a = - 10 - b$

Simplify the right side.

Simplify $(- 10 - b) (3) + b$ .

Simplify each term.

Apply the distributive property.

$- 40 = - 10 \cdot 3 - b \cdot 3 + b$

$- 20 = - 20 - b$

$a = - 10 - b$

Multiply $- 10$ by $3$ .

$- 40 = - 30 - b \cdot 3 + b$

$- 20 = - 20 - b$

$a = - 10 - b$

Multiply $3$ by $- 1$ .

$- 40 = - 30 - 3 b + b$

$- 20 = - 20 - b$

$a = - 10 - b$

$- 40 = - 30 - 3 b + b$

$- 20 = - 20 - b$

$a = - 10 - b$

Add $- 3 b$ and $b$ .

$- 40 = - 30 - 2 b$

$- 20 = - 20 - b$

$a = - 10 - b$

$- 40 = - 30 - 2 b$

$- 20 = - 20 - b$

$a = - 10 - b$

$- 40 = - 30 - 2 b$

$- 20 = - 20 - b$

$a = - 10 - b$

$- 40 = - 30 - 2 b$

$- 20 = - 20 - b$

$a = - 10 - b$

Solve for $b$ in $- 40 = - 30 - 2 b$ .

Rewrite the equation as $- 30 - 2 b = - 40$ .

$- 30 - 2 b = - 40$

$- 20 = - 20 - b$

$a = - 10 - b$

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

Add $30$ to both sides of the equation.

$- 2 b = - 40 + 30$

$- 20 = - 20 - b$

$a = - 10 - b$

Add $- 40$ and $30$ .

$- 2 b = - 10$

$- 20 = - 20 - b$

$a = - 10 - b$

$- 2 b = - 10$

$- 20 = - 20 - b$

$a = - 10 - b$

Divide each term in $- 2 b = - 10$ by $- 2$ and simplify.

Divide each term in $- 2 b = - 10$ by $- 2$ .

$\frac{- 2 b}{- 2} = \frac{- 10}{- 2}$

$- 20 = - 20 - b$

$a = - 10 - b$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 b}{- 2} = \frac{- 10}{- 2}$

$- 20 = - 20 - b$

$a = - 10 - b$

Divide $b$ by $1$ .

$b = \frac{- 10}{- 2}$

$- 20 = - 20 - b$

$a = - 10 - b$

$b = \frac{- 10}{- 2}$

$- 20 = - 20 - b$

$a = - 10 - b$

$b = \frac{- 10}{- 2}$

$- 20 = - 20 - b$

$a = - 10 - b$

Simplify the right side.

Divide $- 10$ by $- 2$ .

$b = 5$

$- 20 = - 20 - b$

$a = - 10 - b$

$b = 5$

$- 20 = - 20 - b$

$a = - 10 - b$

$b = 5$

$- 20 = - 20 - b$

$a = - 10 - b$

$b = 5$

$- 20 = - 20 - b$

$a = - 10 - b$

Replace all occurrences of $b$ with $5$ in each equation.

Replace all occurrences of $b$ in $- 20 = - 20 - b$ with $5$ .

$- 20 = - 20 - (5)$

$b = 5$

$a = - 10 - b$

Simplify the right side.

Simplify $- 20 - (5)$ .

Multiply $- 1$ by $5$ .

$- 20 = - 20 - 5$

$b = 5$

$a = - 10 - b$

Subtract $5$ from $- 20$ .

$- 20 = - 25$

$b = 5$

$a = - 10 - b$

$- 20 = - 25$

$b = 5$

$a = - 10 - b$

$- 20 = - 25$

$b = 5$

$a = - 10 - b$

Replace all occurrences of $b$ in $a = - 10 - b$ with $5$ .

$a = - 10 - (5)$

$- 20 = - 25$

$b = 5$

Simplify the right side.

Simplify $- 10 - (5)$ .

Multiply $- 1$ by $5$ .

$a = - 10 - 5$

$- 20 = - 25$

$b = 5$

Subtract $5$ from $- 10$ .

$a = - 15$

$- 20 = - 25$

$b = 5$

$a = - 15$

$- 20 = - 25$

$b = 5$

$a = - 15$

$- 20 = - 25$

$b = 5$

$a = - 15$

$- 20 = - 25$

$b = 5$

Since $- 20 = - 25$ 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

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 $- 10 = a + b + c$ .

Rewrite the equation as $a + b + c = - 10$ .

$a + b + c = - 10$

$- 20 = a \cdot 2^{2} + b (2) + c$

$- 40 = a \cdot 3^{2} + b (3) + c$

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

Subtract $b$ from both sides of the equation.

$a + c = - 10 - b$

$- 20 = a \cdot 2^{2} + b (2) + c$

$- 40 = a \cdot 3^{2} + b (3) + c$

Subtract $c$ from both sides of the equation.

$a = - 10 - b - c$

$- 20 = a \cdot 2^{2} + b (2) + c$

$- 40 = a \cdot 3^{2} + b (3) + c$

$a = - 10 - b - c$

$- 20 = a \cdot 2^{2} + b (2) + c$

$- 40 = a \cdot 3^{2} + b (3) + c$

$a = - 10 - b - c$

$- 20 = a \cdot 2^{2} + b (2) + c$

$- 40 = a \cdot 3^{2} + b (3) + c$

Replace all occurrences of $a$ with $- 10 - b - c$ in each equation.

Replace all occurrences of $a$ in $- 20 = a \cdot 2^{2} + b (2) + c$ with $- 10 - b - c$ .

$- 20 = (- 10 - b - c) \cdot 2^{2} + b (2) + c$

$a = - 10 - b - c$

$- 40 = a \cdot 3^{2} + b (3) + c$

Simplify the right side.

Simplify $(- 10 - b - c) \cdot 2^{2} + b (2) + c$ .

Simplify each term.

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

$- 20 = (- 10 - b - c) \cdot 4 + b (2) + c$

$a = - 10 - b - c$

$- 40 = a \cdot 3^{2} + b (3) + c$

Apply the distributive property.

$- 20 = - 10 \cdot 4 - b \cdot 4 - c \cdot 4 + b (2) + c$

$a = - 10 - b - c$

$- 40 = a \cdot 3^{2} + b (3) + c$

Simplify.

Multiply $- 10$ by $4$ .

$- 20 = - 40 - b \cdot 4 - c \cdot 4 + b (2) + c$

$a = - 10 - b - c$

$- 40 = a \cdot 3^{2} + b (3) + c$

Multiply $4$ by $- 1$ .

$- 20 = - 40 - 4 b - c \cdot 4 + b (2) + c$

$a = - 10 - b - c$

$- 40 = a \cdot 3^{2} + b (3) + c$

Multiply $4$ by $- 1$ .

$- 20 = - 40 - 4 b - 4 c + b (2) + c$

$a = - 10 - b - c$

$- 40 = a \cdot 3^{2} + b (3) + c$

$- 20 = - 40 - 4 b - 4 c + b (2) + c$

$a = - 10 - b - c$

$- 40 = a \cdot 3^{2} + b (3) + c$

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

$- 20 = - 40 - 4 b - 4 c + 2 b + c$

$a = - 10 - b - c$

$- 40 = a \cdot 3^{2} + b (3) + c$

$- 20 = - 40 - 4 b - 4 c + 2 b + c$

$a = - 10 - b - c$

$- 40 = a \cdot 3^{2} + b (3) + c$

Simplify by adding terms.

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

$- 20 = - 40 - 2 b - 4 c + c$

$a = - 10 - b - c$

$- 40 = a \cdot 3^{2} + b (3) + c$

Add $- 4 c$ and $c$ .

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

$- 40 = a \cdot 3^{2} + b (3) + c$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

$- 40 = a \cdot 3^{2} + b (3) + c$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

$- 40 = a \cdot 3^{2} + b (3) + c$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

$- 40 = a \cdot 3^{2} + b (3) + c$

Replace all occurrences of $a$ in $- 40 = a \cdot 3^{2} + b (3) + c$ with $- 10 - b - c$ .

$- 40 = (- 10 - b - c) \cdot 3^{2} + b (3) + c$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

Simplify the right side.

Simplify $(- 10 - b - c) \cdot 3^{2} + b (3) + c$ .

Simplify each term.

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

$- 40 = (- 10 - b - c) \cdot 9 + b (3) + c$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

Apply the distributive property.

$- 40 = - 10 \cdot 9 - b \cdot 9 - c \cdot 9 + b (3) + c$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

Simplify.

Multiply $- 10$ by $9$ .

$- 40 = - 90 - b \cdot 9 - c \cdot 9 + b (3) + c$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

Multiply $9$ by $- 1$ .

$- 40 = - 90 - 9 b - c \cdot 9 + b (3) + c$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

Multiply $9$ by $- 1$ .

$- 40 = - 90 - 9 b - 9 c + b (3) + c$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

$- 40 = - 90 - 9 b - 9 c + b (3) + c$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

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

$- 40 = - 90 - 9 b - 9 c + 3 b + c$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

$- 40 = - 90 - 9 b - 9 c + 3 b + c$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

Simplify by adding terms.

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

$- 40 = - 90 - 6 b - 9 c + c$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

Add $- 9 c$ and $c$ .

$- 40 = - 90 - 6 b - 8 c$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

$- 40 = - 90 - 6 b - 8 c$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

$- 40 = - 90 - 6 b - 8 c$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

$- 40 = - 90 - 6 b - 8 c$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

$- 40 = - 90 - 6 b - 8 c$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

Solve for $b$ in $- 40 = - 90 - 6 b - 8 c$ .

Rewrite the equation as $- 90 - 6 b - 8 c = - 40$ .

$- 90 - 6 b - 8 c = - 40$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

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

Add $90$ to both sides of the equation.

$- 6 b - 8 c = - 40 + 90$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

Add $8 c$ to both sides of the equation.

$- 6 b = - 40 + 90 + 8 c$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

Add $- 40$ and $90$ .

$- 6 b = 50 + 8 c$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

$- 6 b = 50 + 8 c$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

Divide each term in $- 6 b = 50 + 8 c$ by $- 6$ and simplify.

Divide each term in $- 6 b = 50 + 8 c$ by $- 6$ .

$\frac{- 6 b}{- 6} = \frac{50}{- 6} + \frac{8 c}{- 6}$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

$\frac{- 6 b}{- 6} = \frac{50}{- 6} + \frac{8 c}{- 6}$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

Divide $b$ by $1$ .

$b = \frac{50}{- 6} + \frac{8 c}{- 6}$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

$b = \frac{50}{- 6} + \frac{8 c}{- 6}$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

$b = \frac{50}{- 6} + \frac{8 c}{- 6}$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

Simplify the right side.

Simplify each term.

Cancel the common factor of $50$ and $- 6$ .

Factor $2$ out of $50$ .

$b = \frac{2 (25)}{- 6} + \frac{8 c}{- 6}$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

Cancel the common factors.

Factor $2$ out of $- 6$ .

$b = \frac{2 \cdot 25}{2 \cdot - 3} + \frac{8 c}{- 6}$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

Cancel the common factor.

$b = \frac{2 \cdot 25}{2 \cdot - 3} + \frac{8 c}{- 6}$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

Rewrite the expression.

$b = \frac{25}{- 3} + \frac{8 c}{- 6}$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

$b = \frac{25}{- 3} + \frac{8 c}{- 6}$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

$b = \frac{25}{- 3} + \frac{8 c}{- 6}$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

Move the negative in front of the fraction.

$b = - \frac{25}{3} + \frac{8 c}{- 6}$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

Cancel the common factor of $8$ and $- 6$ .

Factor $2$ out of $8 c$ .

$b = - \frac{25}{3} + \frac{2 (4 c)}{- 6}$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

Cancel the common factors.

Factor $2$ out of $- 6$ .

$b = - \frac{25}{3} + \frac{2 (4 c)}{2 (- 3)}$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

Cancel the common factor.

$b = - \frac{25}{3} + \frac{2 (4 c)}{2 \cdot - 3}$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

Rewrite the expression.

$b = - \frac{25}{3} + \frac{4 c}{- 3}$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

$b = - \frac{25}{3} + \frac{4 c}{- 3}$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

$b = - \frac{25}{3} + \frac{4 c}{- 3}$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

Move the negative in front of the fraction.

$b = - \frac{25}{3} - \frac{4 c}{3}$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$- 20 = - 40 - 2 b - 3 c$

$a = - 10 - b - c$

Replace all occurrences of $b$ with $- \frac{25}{3} - \frac{4 c}{3}$ in each equation.

Replace all occurrences of $b$ in $- 20 = - 40 - 2 b - 3 c$ with $- \frac{25}{3} - \frac{4 c}{3}$ .

$- 20 = - 40 - 2 (- \frac{25}{3} - \frac{4 c}{3}) - 3 c$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 10 - b - c$

Simplify the right side.

Simplify $- 40 - 2 (- \frac{25}{3} - \frac{4 c}{3}) - 3 c$ .

Simplify each term.

Apply the distributive property.

$- 20 = - 40 - 2 (- \frac{25}{3}) - 2 (- \frac{4 c}{3}) - 3 c$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 10 - b - c$

Multiply $- 2 (- \frac{25}{3})$ .

Multiply $- 1$ by $- 2$ .

$- 20 = - 40 + 2 (\frac{25}{3}) - 2 (- \frac{4 c}{3}) - 3 c$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 10 - b - c$

Combine $2$ and $\frac{25}{3}$ .

$- 20 = - 40 + \frac{2 \cdot 25}{3} - 2 (- \frac{4 c}{3}) - 3 c$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 10 - b - c$

Multiply $2$ by $25$ .

$- 20 = - 40 + \frac{50}{3} - 2 (- \frac{4 c}{3}) - 3 c$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 10 - b - c$

$- 20 = - 40 + \frac{50}{3} - 2 (- \frac{4 c}{3}) - 3 c$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 10 - b - c$

Multiply $- 2 (- \frac{4 c}{3})$ .

Multiply $- 1$ by $- 2$ .

$- 20 = - 40 + \frac{50}{3} + 2 (\frac{4 c}{3}) - 3 c$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 10 - b - c$

Combine $2$ and $\frac{4 c}{3}$ .

$- 20 = - 40 + \frac{50}{3} + \frac{2 (4 c)}{3} - 3 c$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 10 - b - c$

Multiply $4$ by $2$ .

$- 20 = - 40 + \frac{50}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 10 - b - c$

$- 20 = - 40 + \frac{50}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 10 - b - c$

$- 20 = - 40 + \frac{50}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 10 - b - c$

To write $- 40$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- 20 = - 40 \cdot \frac{3}{3} + \frac{50}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 10 - b - c$

Combine $- 40$ and $\frac{3}{3}$ .

$- 20 = \frac{- 40 \cdot 3}{3} + \frac{50}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 10 - b - c$

Combine the numerators over the common denominator.

$- 20 = \frac{- 40 \cdot 3 + 50}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 10 - b - c$

Simplify the numerator.

Multiply $- 40$ by $3$ .

$- 20 = \frac{- 120 + 50}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 10 - b - c$

Add $- 120$ and $50$ .

$- 20 = \frac{- 70}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 10 - b - c$

$- 20 = \frac{- 70}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 10 - b - c$

Move the negative in front of the fraction.

$- 20 = - \frac{70}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 10 - b - c$

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

$- 20 = - \frac{70}{3} + \frac{8 c}{3} - 3 c \cdot \frac{3}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 10 - b - c$

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

$- 20 = - \frac{70}{3} + \frac{8 c}{3} + \frac{- 3 c \cdot 3}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 10 - b - c$

Combine the numerators over the common denominator.

$- 20 = - \frac{70}{3} + \frac{8 c - 3 c \cdot 3}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 10 - b - c$

Combine the numerators over the common denominator.

$- 20 = \frac{- 70 + 8 c - 3 c \cdot 3}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 10 - b - c$

Multiply $3$ by $- 3$ .

$- 20 = \frac{- 70 + 8 c - 9 c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 10 - b - c$

Subtract $9 c$ from $8 c$ .

$- 20 = \frac{- 70 - c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 10 - b - c$

Rewrite $- 70$ as $- 1 (70)$ .

$- 20 = \frac{- 1 \cdot 70 - c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 10 - b - c$

Factor $- 1$ out of $- c$ .

$- 20 = \frac{- 1 \cdot 70 - (c)}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 10 - b - c$

Factor $- 1$ out of $- 1 (70) - (c)$ .

$- 20 = \frac{- 1 (70 + c)}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 10 - b - c$

Move the negative in front of the fraction.

$- 20 = - \frac{70 + c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 10 - b - c$

$- 20 = - \frac{70 + c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 10 - b - c$

$- 20 = - \frac{70 + c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 10 - b - c$

Replace all occurrences of $b$ in $a = - 10 - b - c$ with $- \frac{25}{3} - \frac{4 c}{3}$ .

$a = - 10 - (- \frac{25}{3} - \frac{4 c}{3}) - c$

$- 20 = - \frac{70 + c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

Simplify the right side.

Simplify $- 10 - (- \frac{25}{3} - \frac{4 c}{3}) - c$ .

Simplify each term.

Apply the distributive property.

$a = - 10 + \frac{25}{3} + \frac{4 c}{3} - c$

$- 20 = - \frac{70 + c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

Multiply $- - \frac{25}{3}$ .

Multiply $- 1$ by $- 1$ .

$a = - 10 + 1 (\frac{25}{3}) + \frac{4 c}{3} - c$

$- 20 = - \frac{70 + c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

Multiply $\frac{25}{3}$ by $1$ .

$a = - 10 + \frac{25}{3} + \frac{4 c}{3} - c$

$- 20 = - \frac{70 + c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 10 + \frac{25}{3} + \frac{4 c}{3} - c$

$- 20 = - \frac{70 + c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

Multiply $- - \frac{4 c}{3}$ .

Multiply $- 1$ by $- 1$ .

$a = - 10 + \frac{25}{3} + 1 (\frac{4 c}{3}) - c$

$- 20 = - \frac{70 + c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

Multiply $\frac{4 c}{3}$ by $1$ .

$a = - 10 + \frac{25}{3} + \frac{4 c}{3} - c$

$- 20 = - \frac{70 + c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 10 + \frac{25}{3} + \frac{4 c}{3} - c$

$- 20 = - \frac{70 + c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 10 + \frac{25}{3} + \frac{4 c}{3} - c$

$- 20 = - \frac{70 + c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

To write $- 10$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$a = - 10 \cdot \frac{3}{3} + \frac{25}{3} + \frac{4 c}{3} - c$

$- 20 = - \frac{70 + c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

Combine $- 10$ and $\frac{3}{3}$ .

$a = \frac{- 10 \cdot 3}{3} + \frac{25}{3} + \frac{4 c}{3} - c$

$- 20 = - \frac{70 + c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

Combine the numerators over the common denominator.

$a = \frac{- 10 \cdot 3 + 25}{3} + \frac{4 c}{3} - c$

$- 20 = - \frac{70 + c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

Simplify the numerator.

Multiply $- 10$ by $3$ .

$a = \frac{- 30 + 25}{3} + \frac{4 c}{3} - c$

$- 20 = - \frac{70 + c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

Add $- 30$ and $25$ .

$a = \frac{- 5}{3} + \frac{4 c}{3} - c$

$- 20 = - \frac{70 + c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = \frac{- 5}{3} + \frac{4 c}{3} - c$

$- 20 = - \frac{70 + c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

Move the negative in front of the fraction.

$a = - \frac{5}{3} + \frac{4 c}{3} - c$

$- 20 = - \frac{70 + c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

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

$a = - \frac{5}{3} + \frac{4 c}{3} - c \cdot \frac{3}{3}$

$- 20 = - \frac{70 + c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

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

$a = - \frac{5}{3} + \frac{4 c}{3} + \frac{- c \cdot 3}{3}$

$- 20 = - \frac{70 + c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

Combine the numerators over the common denominator.

$a = - \frac{5}{3} + \frac{4 c - c \cdot 3}{3}$

$- 20 = - \frac{70 + c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

Combine the numerators over the common denominator.

$a = \frac{- 5 + 4 c - c \cdot 3}{3}$

$- 20 = - \frac{70 + c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

Multiply $3$ by $- 1$ .

$a = \frac{- 5 + 4 c - 3 c}{3}$

$- 20 = - \frac{70 + c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

Subtract $3 c$ from $4 c$ .

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

$- 20 = - \frac{70 + c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

Rewrite $- 5$ as $- 1 (5)$ .

$a = \frac{- 1 \cdot 5 + c}{3}$

$- 20 = - \frac{70 + c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

Factor $- 1$ out of $c$ .

$a = \frac{- 1 \cdot 5 - 1 (- c)}{3}$

$- 20 = - \frac{70 + c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

Factor $- 1$ out of $- 1 (5) - 1 (- c)$ .

$a = \frac{- 1 (5 - c)}{3}$

$- 20 = - \frac{70 + c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

Move the negative in front of the fraction.

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

$- 20 = - \frac{70 + c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

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

$- 20 = - \frac{70 + c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

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

$- 20 = - \frac{70 + c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

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

$- 20 = - \frac{70 + c}{3}$

$b = - \frac{25}{3} - \frac{4 c}{3}$

Solve for $c$ in $- 20 = - \frac{70 + c}{3}$ .

Rewrite the equation as $- \frac{70 + c}{3} = - 20$ .

$- \frac{70 + c}{3} = - 20$

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

$b = - \frac{25}{3} - \frac{4 c}{3}$

Multiply both sides of the equation by $- 3$ .

$- 3 (- \frac{70 + c}{3}) = - 3 \cdot - 20$

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

$b = - \frac{25}{3} - \frac{4 c}{3}$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- 3 (- \frac{70 + c}{3})$ .

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{70 + c}{3}$ into the numerator.

$- 3 \frac{- (70 + c)}{3} = - 3 \cdot - 20$

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

$b = - \frac{25}{3} - \frac{4 c}{3}$

Factor $3$ out of $- 3$ .

$3 (- 1) (\frac{- (70 + c)}{3}) = - 3 \cdot - 20$

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

$b = - \frac{25}{3} - \frac{4 c}{3}$

Cancel the common factor.

$3 \cdot (- 1 \frac{- (70 + c)}{3}) = - 3 \cdot - 20$

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

$b = - \frac{25}{3} - \frac{4 c}{3}$

Rewrite the expression.

$70 + c = - 3 \cdot - 20$

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

$b = - \frac{25}{3} - \frac{4 c}{3}$

$70 + c = - 3 \cdot - 20$

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

$b = - \frac{25}{3} - \frac{4 c}{3}$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 (70 + c) = - 3 \cdot - 20$

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

$b = - \frac{25}{3} - \frac{4 c}{3}$

Multiply $70 + c$ by $1$ .

$70 + c = - 3 \cdot - 20$

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

$b = - \frac{25}{3} - \frac{4 c}{3}$

$70 + c = - 3 \cdot - 20$

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

$b = - \frac{25}{3} - \frac{4 c}{3}$

$70 + c = - 3 \cdot - 20$

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

$b = - \frac{25}{3} - \frac{4 c}{3}$

$70 + c = - 3 \cdot - 20$

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

$b = - \frac{25}{3} - \frac{4 c}{3}$

Simplify the right side.

Multiply $- 3$ by $- 20$ .

$70 + c = 60$

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

$b = - \frac{25}{3} - \frac{4 c}{3}$

$70 + c = 60$

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

$b = - \frac{25}{3} - \frac{4 c}{3}$

$70 + c = 60$

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

$b = - \frac{25}{3} - \frac{4 c}{3}$

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

Subtract $70$ from both sides of the equation.

$c = 60 - 70$

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

$b = - \frac{25}{3} - \frac{4 c}{3}$

Subtract $70$ from $60$ .

$c = - 10$

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

$b = - \frac{25}{3} - \frac{4 c}{3}$

$c = - 10$

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

$b = - \frac{25}{3} - \frac{4 c}{3}$

$c = - 10$

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

$b = - \frac{25}{3} - \frac{4 c}{3}$

Replace all occurrences of $c$ with $- 10$ in each equation.

Replace all occurrences of $c$ in $a = - \frac{5 - c}{3}$ with $- 10$ .

$a = - \frac{5 - (- 10)}{3}$

$c = - 10$

$b = - \frac{25}{3} - \frac{4 c}{3}$

Simplify the right side.

Simplify $- \frac{5 - (- 10)}{3}$ .

Simplify the numerator.

Multiply $- 1$ by $- 10$ .

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

$c = - 10$

$b = - \frac{25}{3} - \frac{4 c}{3}$

Add $5$ and $10$ .

$a = - \frac{15}{3}$

$c = - 10$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - \frac{15}{3}$

$c = - 10$

$b = - \frac{25}{3} - \frac{4 c}{3}$

Simplify the expression.

Divide $15$ by $3$ .

$a = - 1 \cdot 5$

$c = - 10$

$b = - \frac{25}{3} - \frac{4 c}{3}$

Multiply $- 1$ by $5$ .

$a = - 5$

$c = - 10$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 5$

$c = - 10$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 5$

$c = - 10$

$b = - \frac{25}{3} - \frac{4 c}{3}$

$a = - 5$

$c = - 10$

$b = - \frac{25}{3} - \frac{4 c}{3}$

Replace all occurrences of $c$ in $b = - \frac{25}{3} - \frac{4 c}{3}$ with $- 10$ .

$b = - \frac{25}{3} - \frac{4 (- 10)}{3}$

$a = - 5$

$c = - 10$

Simplify the right side.

Simplify $- \frac{25}{3} - \frac{4 (- 10)}{3}$ .

Combine the numerators over the common denominator.

$b = \frac{- 25 - 4 \cdot - 10}{3}$

$a = - 5$

$c = - 10$

Simplify the expression.

Multiply $- 4$ by $- 10$ .

$b = \frac{- 25 + 40}{3}$

$a = - 5$

$c = - 10$

Add $- 25$ and $40$ .

$b = \frac{15}{3}$

$a = - 5$

$c = - 10$

Divide $15$ by $3$ .

$b = 5$

$a = - 5$

$c = - 10$

$b = 5$

$a = - 5$

$c = - 10$

$b = 5$

$a = - 5$

$c = - 10$

$b = 5$

$a = - 5$

$c = - 10$

$b = 5$

$a = - 5$

$c = - 10$

List all of the solutions.

$b = 5, a = - 5, c = - 10$

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 $a = - 5$ , $b = 5$ , $c = - 10$ , and $x = 1$ .

Simplify each term.

One to any power is one.

$y = - 5 \cdot 1 + (5) \cdot (1) - 10$

Multiply $- 5$ by $1$ .

$y = - 5 + (5) \cdot (1) - 10$

Multiply $5$ by $1$ .

$y = - 5 + 5 - 10$

Simplify by adding and subtracting.

Add $- 5$ and $5$ .

$y = 0 - 10$

Subtract $10$ from $0$ .

$y = - 10$

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 1$ . This check passes since $y = - 10$ and $y = - 10$ .

$- 10 = - 10$

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = - 5$ , $b = 5$ , $c = - 10$ , and $x = 2$ .

Simplify each term.

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

$y = - 5 \cdot 4 + (5) \cdot (2) - 10$

Multiply $- 5$ by $4$ .

$y = - 20 + (5) \cdot (2) - 10$

Multiply $5$ by $2$ .

$y = - 20 + 10 - 10$

Simplify by adding and subtracting.

Add $- 20$ and $10$ .

$y = - 10 - 10$

Subtract $10$ from $- 10$ .

$y = - 20$

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 2$ . This check passes since $y = - 20$ and $y = - 20$ .

$- 20 = - 20$

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = - 5$ , $b = 5$ , $c = - 10$ , and $x = 3$ .

Simplify each term.

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

$y = - 5 \cdot 9 + (5) \cdot (3) - 10$

Multiply $- 5$ by $9$ .

$y = - 45 + (5) \cdot (3) - 10$

Multiply $5$ by $3$ .

$y = - 45 + 15 - 10$

Simplify by adding and subtracting.

Add $- 45$ and $15$ .

$y = - 30 - 10$

Subtract $10$ from $- 30$ .

$y = - 40$

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 3$ . This check passes since $y = - 40$ and $y = - 40$ .

$- 40 = - 40$

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

The function is quadratic

Since all $y = y$ , the function is quadratic and follows the form $y = - 5 x^{2} + 5 x - 10$ .

$y = - 5 x^{2} + 5 x - 10$

Find the Function Rule table[[x,y],[1,-10],[2,-20],[3,-40]]

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