Find the Function Rule table[[x,y],[1,-7],[2,-10],[3,-13]]

$\begin{array}{c} x & y \\ 1 & - 7 \\ 2 & - 10 \\ 3 & - 13 \end{array}$

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 $y = a x + b$ .

$- 7 = a (1) + b - 10 = a (2) + b - 13 = a (3) + b$

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

Solve for $a$ in $- 7 = a + b$ .

Rewrite the equation as $a + b = - 7$ .

$a + b = - 7$

$- 10 = a (2) + b$

$- 13 = a (3) + b$

Subtract $b$ from both sides of the equation.

$a = - 7 - b$

$- 10 = a (2) + b$

$- 13 = a (3) + b$

$a = - 7 - b$

$- 10 = a (2) + b$

$- 13 = a (3) + b$

Replace all occurrences of $a$ with $- 7 - b$ in each equation.

Replace all occurrences of $a$ in $- 10 = a (2) + b$ with $- 7 - b$ .

$- 10 = (- 7 - b) (2) + b$

$a = - 7 - b$

$- 13 = a (3) + b$

Simplify the right side.

Simplify $(- 7 - b) (2) + b$ .

Simplify each term.

Apply the distributive property.

$- 10 = - 7 \cdot 2 - b \cdot 2 + b$

$a = - 7 - b$

$- 13 = a (3) + b$

Multiply $- 7$ by $2$ .

$- 10 = - 14 - b \cdot 2 + b$

$a = - 7 - b$

$- 13 = a (3) + b$

Multiply $2$ by $- 1$ .

$- 10 = - 14 - 2 b + b$

$a = - 7 - b$

$- 13 = a (3) + b$

$- 10 = - 14 - 2 b + b$

$a = - 7 - b$

$- 13 = a (3) + b$

Add $- 2 b$ and $b$ .

$- 10 = - 14 - b$

$a = - 7 - b$

$- 13 = a (3) + b$

$- 10 = - 14 - b$

$a = - 7 - b$

$- 13 = a (3) + b$

$- 10 = - 14 - b$

$a = - 7 - b$

$- 13 = a (3) + b$

Replace all occurrences of $a$ in $- 13 = a (3) + b$ with $- 7 - b$ .

$- 13 = (- 7 - b) (3) + b$

$- 10 = - 14 - b$

$a = - 7 - b$

Simplify the right side.

Simplify $(- 7 - b) (3) + b$ .

Simplify each term.

Apply the distributive property.

$- 13 = - 7 \cdot 3 - b \cdot 3 + b$

$- 10 = - 14 - b$

$a = - 7 - b$

Multiply $- 7$ by $3$ .

$- 13 = - 21 - b \cdot 3 + b$

$- 10 = - 14 - b$

$a = - 7 - b$

Multiply $3$ by $- 1$ .

$- 13 = - 21 - 3 b + b$

$- 10 = - 14 - b$

$a = - 7 - b$

$- 13 = - 21 - 3 b + b$

$- 10 = - 14 - b$

$a = - 7 - b$

Add $- 3 b$ and $b$ .

$- 13 = - 21 - 2 b$

$- 10 = - 14 - b$

$a = - 7 - b$

$- 13 = - 21 - 2 b$

$- 10 = - 14 - b$

$a = - 7 - b$

$- 13 = - 21 - 2 b$

$- 10 = - 14 - b$

$a = - 7 - b$

$- 13 = - 21 - 2 b$

$- 10 = - 14 - b$

$a = - 7 - b$

Solve for $b$ in $- 13 = - 21 - 2 b$ .

Rewrite the equation as $- 21 - 2 b = - 13$ .

$- 21 - 2 b = - 13$

$- 10 = - 14 - b$

$a = - 7 - b$

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

Add $21$ to both sides of the equation.

$- 2 b = - 13 + 21$

$- 10 = - 14 - b$

$a = - 7 - b$

Add $- 13$ and $21$ .

$- 2 b = 8$

$- 10 = - 14 - b$

$a = - 7 - b$

$- 2 b = 8$

$- 10 = - 14 - b$

$a = - 7 - b$

Divide each term in $- 2 b = 8$ by $- 2$ and simplify.

Divide each term in $- 2 b = 8$ by $- 2$ .

$\frac{- 2 b}{- 2} = \frac{8}{- 2}$

$- 10 = - 14 - b$

$a = - 7 - b$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 b}{- 2} = \frac{8}{- 2}$

$- 10 = - 14 - b$

$a = - 7 - b$

Divide $b$ by $1$ .

$b = \frac{8}{- 2}$

$- 10 = - 14 - b$

$a = - 7 - b$

$b = \frac{8}{- 2}$

$- 10 = - 14 - b$

$a = - 7 - b$

$b = \frac{8}{- 2}$

$- 10 = - 14 - b$

$a = - 7 - b$

Simplify the right side.

Divide $8$ by $- 2$ .

$b = - 4$

$- 10 = - 14 - b$

$a = - 7 - b$

$b = - 4$

$- 10 = - 14 - b$

$a = - 7 - b$

$b = - 4$

$- 10 = - 14 - b$

$a = - 7 - b$

$b = - 4$

$- 10 = - 14 - b$

$a = - 7 - b$

Replace all occurrences of $b$ with $- 4$ in each equation.

Replace all occurrences of $b$ in $- 10 = - 14 - b$ with $- 4$ .

$- 10 = - 14 - (- 4)$

$b = - 4$

$a = - 7 - b$

Simplify the right side.

Simplify $- 14 - (- 4)$ .

Multiply $- 1$ by $- 4$ .

$- 10 = - 14 + 4$

$b = - 4$

$a = - 7 - b$

Add $- 14$ and $4$ .

$- 10 = - 10$

$b = - 4$

$a = - 7 - b$

$- 10 = - 10$

$b = - 4$

$a = - 7 - b$

$- 10 = - 10$

$b = - 4$

$a = - 7 - b$

Replace all occurrences of $b$ in $a = - 7 - b$ with $- 4$ .

$a = - 7 - (- 4)$

$- 10 = - 10$

$b = - 4$

Simplify the right side.

Simplify $- 7 - (- 4)$ .

Multiply $- 1$ by $- 4$ .

$a = - 7 + 4$

$- 10 = - 10$

$b = - 4$

Add $- 7$ and $4$ .

$a = - 3$

$- 10 = - 10$

$b = - 4$

$a = - 3$

$- 10 = - 10$

$b = - 4$

$a = - 3$

$- 10 = - 10$

$b = - 4$

$a = - 3$

$- 10 = - 10$

$b = - 4$

Remove any equations from the system that are always true.

$a = - 3$

$b = - 4$

List all of the solutions.

$a = - 3, b = - 4$

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 $a = - 3$ , $b = - 4$ , and $x = 1$ .

Multiply $- 3$ by $1$ .

$y = - 3 - 4$

Subtract $4$ from $- 3$ .

$y = - 7$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 1$ . This check passes since $y = - 7$ and $y = - 7$ .

$- 7 = - 7$

Calculate the value of $y$ when $a = - 3$ , $b = - 4$ , and $x = 2$ .

Multiply $- 3$ by $2$ .

$y = - 6 - 4$

Subtract $4$ from $- 6$ .

$y = - 10$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 2$ . This check passes since $y = - 10$ and $y = - 10$ .

$- 10 = - 10$

Calculate the value of $y$ when $a = - 3$ , $b = - 4$ , and $x = 3$ .

Multiply $- 3$ by $3$ .

$y = - 9 - 4$

Subtract $4$ from $- 9$ .

$y = - 13$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 3$ . This check passes since $y = - 13$ and $y = - 13$ .

$- 13 = - 13$

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

The function is linear

Since all $y = y$ , the function is linear and follows the form $y = - 3 x - 4$ .

$y = - 3 x - 4$

Find the Function Rule table[[x,y],[1,-7],[2,-10],[3,-13]]

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