Find the Function Rule table[[x,y],[-20,-268],[-14,-196],[-8,-124]]

$\begin{array}{c} x & y \\ - 20 & - 268 \\ - 14 & - 196 \\ - 8 & - 124 \end{array}$

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 $y = a x + b$ .

$- 268 = a (- 20) + b - 196 = a (- 14) + b - 124 = a (- 8) + b$

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

Solve for $b$ in $- 268 = a (- 20) + b$ .

Rewrite the equation as $a (- 20) + b = - 268$ .

$a (- 20) + b = - 268$

$- 196 = a (- 14) + b$

$- 124 = a (- 8) + b$

Move $- 20$ to the left of $a$ .

$- 20 a + b = - 268$

$- 196 = a (- 14) + b$

$- 124 = a (- 8) + b$

Add $20 a$ to both sides of the equation.

$b = - 268 + 20 a$

$- 196 = a (- 14) + b$

$- 124 = a (- 8) + b$

$b = - 268 + 20 a$

$- 196 = a (- 14) + b$

$- 124 = a (- 8) + b$

Replace all occurrences of $b$ with $- 268 + 20 a$ in each equation.

Replace all occurrences of $b$ in $- 196 = a (- 14) + b$ with $- 268 + 20 a$ .

$- 196 = a (- 14) - 268 + 20 a$

$b = - 268 + 20 a$

$- 124 = a (- 8) + b$

Simplify $- 196 = a (- 14) - 268 + 20 a$ .

Simplify the left side.

Remove parentheses.

$- 196 = a (- 14) - 268 + 20 a$

$b = - 268 + 20 a$

$- 124 = a (- 8) + b$

$- 196 = a (- 14) - 268 + 20 a$

$b = - 268 + 20 a$

$- 124 = a (- 8) + b$

Simplify the right side.

Simplify $a (- 14) - 268 + 20 a$ .

Move $- 14$ to the left of $a$ .

$- 196 = - 14 a - 268 + 20 a$

$b = - 268 + 20 a$

$- 124 = a (- 8) + b$

Add $- 14 a$ and $20 a$ .

$- 196 = 6 a - 268$

$b = - 268 + 20 a$

$- 124 = a (- 8) + b$

$- 196 = 6 a - 268$

$b = - 268 + 20 a$

$- 124 = a (- 8) + b$

$- 196 = 6 a - 268$

$b = - 268 + 20 a$

$- 124 = a (- 8) + b$

$- 196 = 6 a - 268$

$b = - 268 + 20 a$

$- 124 = a (- 8) + b$

Replace all occurrences of $b$ in $- 124 = a (- 8) + b$ with $- 268 + 20 a$ .

$- 124 = a (- 8) - 268 + 20 a$

$- 196 = 6 a - 268$

$b = - 268 + 20 a$

Simplify $- 124 = a (- 8) - 268 + 20 a$ .

Simplify the left side.

Remove parentheses.

$- 124 = a (- 8) - 268 + 20 a$

$- 196 = 6 a - 268$

$b = - 268 + 20 a$

$- 124 = a (- 8) - 268 + 20 a$

$- 196 = 6 a - 268$

$b = - 268 + 20 a$

Simplify the right side.

Simplify $a (- 8) - 268 + 20 a$ .

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

$- 124 = - 8 a - 268 + 20 a$

$- 196 = 6 a - 268$

$b = - 268 + 20 a$

Add $- 8 a$ and $20 a$ .

$- 124 = 12 a - 268$

$- 196 = 6 a - 268$

$b = - 268 + 20 a$

$- 124 = 12 a - 268$

$- 196 = 6 a - 268$

$b = - 268 + 20 a$

$- 124 = 12 a - 268$

$- 196 = 6 a - 268$

$b = - 268 + 20 a$

$- 124 = 12 a - 268$

$- 196 = 6 a - 268$

$b = - 268 + 20 a$

$- 124 = 12 a - 268$

$- 196 = 6 a - 268$

$b = - 268 + 20 a$

Solve for $a$ in $- 124 = 12 a - 268$ .

Rewrite the equation as $12 a - 268 = - 124$ .

$12 a - 268 = - 124$

$- 196 = 6 a - 268$

$b = - 268 + 20 a$

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

Add $268$ to both sides of the equation.

$12 a = - 124 + 268$

$- 196 = 6 a - 268$

$b = - 268 + 20 a$

Add $- 124$ and $268$ .

$12 a = 144$

$- 196 = 6 a - 268$

$b = - 268 + 20 a$

$12 a = 144$

$- 196 = 6 a - 268$

$b = - 268 + 20 a$

Divide each term in $12 a = 144$ by $12$ and simplify.

Divide each term in $12 a = 144$ by $12$ .

$\frac{12 a}{12} = \frac{144}{12}$

$- 196 = 6 a - 268$

$b = - 268 + 20 a$

Simplify the left side.

Cancel the common factor of $12$ .

Cancel the common factor.

$\frac{12 a}{12} = \frac{144}{12}$

$- 196 = 6 a - 268$

$b = - 268 + 20 a$

Divide $a$ by $1$ .

$a = \frac{144}{12}$

$- 196 = 6 a - 268$

$b = - 268 + 20 a$

$a = \frac{144}{12}$

$- 196 = 6 a - 268$

$b = - 268 + 20 a$

$a = \frac{144}{12}$

$- 196 = 6 a - 268$

$b = - 268 + 20 a$

Simplify the right side.

Divide $144$ by $12$ .

$a = 12$

$- 196 = 6 a - 268$

$b = - 268 + 20 a$

$a = 12$

$- 196 = 6 a - 268$

$b = - 268 + 20 a$

$a = 12$

$- 196 = 6 a - 268$

$b = - 268 + 20 a$

$a = 12$

$- 196 = 6 a - 268$

$b = - 268 + 20 a$

Replace all occurrences of $a$ with $12$ in each equation.

Replace all occurrences of $a$ in $- 196 = 6 a - 268$ with $12$ .

$- 196 = 6 (12) - 268$

$a = 12$

$b = - 268 + 20 a$

Simplify the right side.

Simplify $6 (12) - 268$ .

Multiply $6$ by $12$ .

$- 196 = 72 - 268$

$a = 12$

$b = - 268 + 20 a$

Subtract $268$ from $72$ .

$- 196 = - 196$

$a = 12$

$b = - 268 + 20 a$

$- 196 = - 196$

$a = 12$

$b = - 268 + 20 a$

$- 196 = - 196$

$a = 12$

$b = - 268 + 20 a$

Replace all occurrences of $a$ in $b = - 268 + 20 a$ with $12$ .

$b = - 268 + 20 (12)$

$- 196 = - 196$

$a = 12$

Simplify the right side.

Simplify $- 268 + 20 (12)$ .

Multiply $20$ by $12$ .

$b = - 268 + 240$

$- 196 = - 196$

$a = 12$

Add $- 268$ and $240$ .

$b = - 28$

$- 196 = - 196$

$a = 12$

$b = - 28$

$- 196 = - 196$

$a = 12$

$b = - 28$

$- 196 = - 196$

$a = 12$

$b = - 28$

$- 196 = - 196$

$a = 12$

Remove any equations from the system that are always true.

$b = - 28$

$a = 12$

List all of the solutions.

$b = - 28, a = 12$

Calculate the value of $y$ using each $x$ value in the relation and compare this value to the given $y$ value in the relation.

Calculate the value of $y$ when $a = 12$ , $b = - 28$ , and $x = - 20$ .

Multiply $12$ by $- 20$ .

$y = - 240 - 28$

Subtract $28$ from $- 240$ .

$y = - 268$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = - 20$ . This check passes since $y = - 268$ and $y = - 268$ .

$- 268 = - 268$

Calculate the value of $y$ when $a = 12$ , $b = - 28$ , and $x = - 14$ .

Multiply $12$ by $- 14$ .

$y = - 168 - 28$

Subtract $28$ from $- 168$ .

$y = - 196$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = - 14$ . This check passes since $y = - 196$ and $y = - 196$ .

$- 196 = - 196$

Calculate the value of $y$ when $a = 12$ , $b = - 28$ , and $x = - 8$ .

Multiply $12$ by $- 8$ .

$y = - 96 - 28$

Subtract $28$ from $- 96$ .

$y = - 124$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = - 8$ . This check passes since $y = - 124$ and $y = - 124$ .

$- 124 = - 124$

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

The function is linear

Step 2

Since all $y = y$ , the function is linear and follows the form $y = 12 x - 28$ .

$y = 12 x - 28$

Find the Function Rule table[[x,y],[-20,-268],[-14,-196],[-8,-124]]

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