Find the Function Rule table[[x,y],[0,-10],[1,-7],[2,-4]]

$\begin{array}{c} x & y \\ 0 & - 10 \\ 1 & - 7 \\ 2 & - 4 \end{array}$

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 $y = a x + b$ .

$- 10 = a (0) + b - 7 = a (1) + b - 4 = a (2) + b$

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

Rewrite the equation as $b = - 10$ .

$b = - 10$

$- 7 = a + b$

$- 4 = a (2) + b$

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

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

$- 7 = a - 10$

$b = - 10$

$- 4 = a (2) + b$

Simplify the left side.

Remove parentheses.

$- 7 = a - 10$

$b = - 10$

$- 4 = a (2) + b$

$- 7 = a - 10$

$b = - 10$

$- 4 = a (2) + b$

Replace all occurrences of $b$ in $- 4 = a (2) + b$ with $- 10$ .

$- 4 = a (2) - 10$

$- 7 = a - 10$

$b = - 10$

Simplify $- 4 = a (2) - 10$ .

Simplify the left side.

Remove parentheses.

$- 4 = a (2) - 10$

$- 7 = a - 10$

$b = - 10$

$- 4 = a (2) - 10$

$- 7 = a - 10$

$b = - 10$

Simplify the right side.

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

$- 4 = 2 a - 10$

$- 7 = a - 10$

$b = - 10$

$- 4 = 2 a - 10$

$- 7 = a - 10$

$b = - 10$

$- 4 = 2 a - 10$

$- 7 = a - 10$

$b = - 10$

$- 4 = 2 a - 10$

$- 7 = a - 10$

$b = - 10$

Solve for $a$ in $- 4 = 2 a - 10$ .

Rewrite the equation as $2 a - 10 = - 4$ .

$2 a - 10 = - 4$

$- 7 = a - 10$

$b = - 10$

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

Add $10$ to both sides of the equation.

$2 a = - 4 + 10$

$- 7 = a - 10$

$b = - 10$

Add $- 4$ and $10$ .

$2 a = 6$

$- 7 = a - 10$

$b = - 10$

$2 a = 6$

$- 7 = a - 10$

$b = - 10$

Divide each term in $2 a = 6$ by $2$ and simplify.

Divide each term in $2 a = 6$ by $2$ .

$\frac{2 a}{2} = \frac{6}{2}$

$- 7 = a - 10$

$b = - 10$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 a}{2} = \frac{6}{2}$

$- 7 = a - 10$

$b = - 10$

Divide $a$ by $1$ .

$a = \frac{6}{2}$

$- 7 = a - 10$

$b = - 10$

$a = \frac{6}{2}$

$- 7 = a - 10$

$b = - 10$

$a = \frac{6}{2}$

$- 7 = a - 10$

$b = - 10$

Simplify the right side.

Divide $6$ by $2$ .

$a = 3$

$- 7 = a - 10$

$b = - 10$

$a = 3$

$- 7 = a - 10$

$b = - 10$

$a = 3$

$- 7 = a - 10$

$b = - 10$

$a = 3$

$- 7 = a - 10$

$b = - 10$

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

Replace all occurrences of $a$ in $- 7 = a - 10$ with $3$ .

$- 7 = (3) - 10$

$a = 3$

$b = - 10$

Simplify the right side.

Subtract $10$ from $3$ .

$- 7 = - 7$

$a = 3$

$b = - 10$

$- 7 = - 7$

$a = 3$

$b = - 10$

$- 7 = - 7$

$a = 3$

$b = - 10$

Remove any equations from the system that are always true.

$a = 3$

$b = - 10$

List all of the solutions.

$a = 3, b = - 10$

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 $a = 3$ , $b = - 10$ , and $x = 0$ .

Multiply $3$ by $0$ .

$y = 0 - 10$

Subtract $10$ from $0$ .

$y = - 10$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 0$ . This check passes since $y = - 10$ and $y = - 10$ .

$- 10 = - 10$

Calculate the value of $y$ when $a = 3$ , $b = - 10$ , and $x = 1$ .

Multiply $3$ by $1$ .

$y = 3 - 10$

Subtract $10$ 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 $a = 3$ , $b = - 10$ , and $x = 2$ .

Multiply $3$ by $2$ .

$y = 6 - 10$

Subtract $10$ from $6$ .

$y = - 4$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 2$ . This check passes since $y = - 4$ and $y = - 4$ .

$- 4 = - 4$

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 $y = 3 x - 10$ .

$y = 3 x - 10$

Find the Function Rule table[[x,y],[0,-10],[1,-7],[2,-4]]

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