Examples

[\begin{matrix} 1 & 2 & 7 & 8 & 9 - 2 & - 3 & 1 & 1 & 1 \end{matrix}] = [\begin{matrix} x & 6 & z & 24 & 27 - 6 & - 9 & 3 & 3 & 3 \end{matrix}]

$[\begin{array}{c} 1 & 2 & 7 & 8 & 9 \\ - 2 & - 3 & 1 & 1 & 1 \end{array}] = [\begin{array}{c} x & 6 & z & 24 & 27 \\ - 6 & - 9 & 3 & 3 & 3 \end{array}]$

Step 1

Find the function rule.

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Step 1.1

Check if the function rule is linear.

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Step 1.1.1

To find if the table follows a function rule, check to see if the values follow the linear form

$y = a x + b$ .

$y = a x + b$

Step 1.1.2

Build a set of equations from the table such that

$y = a x + b$ .

$6 = a (2) + b 24 = a (8) + b 27 = a (9) + b - 6 = a (- 2) + b - 9 = a (- 3) + b 3 = a (1) + b 3 = a (1) + b 3 = a (1) + b$

Step 1.1.3

Calculate the values of

$a$ and

$b$ .

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Step 1.1.3.1

Solve for

$a$ in

$3 = a + b$ .

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Step 1.1.3.1.1

Rewrite the equation as

$a + b = 3$ .

$a + b = 3$

$6 = a (2) + b$

$24 = a (8) + b$

$27 = a (9) + b$

$- 6 = a (- 2) + b$

$- 9 = a (- 3) + b$

Step 1.1.3.1.2

Subtract

$b$ from both sides of the equation.

$a = 3 - b$

$6 = a (2) + b$

$24 = a (8) + b$

$27 = a (9) + b$

$- 6 = a (- 2) + b$

$- 9 = a (- 3) + b$

$a = 3 - b$

$6 = a (2) + b$

$24 = a (8) + b$

$27 = a (9) + b$

$- 6 = a (- 2) + b$

$- 9 = a (- 3) + b$

Step 1.1.3.2

Replace all occurrences of

$a$ with

$3 - b$ in each equation.

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Step 1.1.3.2.1

Replace all occurrences of

$a$ in

$6 = a (2) + b$ with

$3 - b$ .

$6 = (3 - b) (2) + b$

$a = 3 - b$

$24 = a (8) + b$

$27 = a (9) + b$

$- 6 = a (- 2) + b$

$- 9 = a (- 3) + b$

Step 1.1.3.2.2

Simplify the right side.

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Step 1.1.3.2.2.1

Simplify

$(3 - b) (2) + b$ .

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Step 1.1.3.2.2.1.1

Simplify each term.

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Step 1.1.3.2.2.1.1.1

Apply the distributive property.

$6 = 3 \cdot 2 - b \cdot 2 + b$

$a = 3 - b$

$24 = a (8) + b$

$27 = a (9) + b$

$- 6 = a (- 2) + b$

$- 9 = a (- 3) + b$

Step 1.1.3.2.2.1.1.2

Multiply

$3$ by

$2$ .

$6 = 6 - b \cdot 2 + b$

$a = 3 - b$

$24 = a (8) + b$

$27 = a (9) + b$

$- 6 = a (- 2) + b$

$- 9 = a (- 3) + b$

Step 1.1.3.2.2.1.1.3

Multiply

$2$ by

$- 1$ .

$6 = 6 - 2 b + b$

$a = 3 - b$

$24 = a (8) + b$

$27 = a (9) + b$

$- 6 = a (- 2) + b$

$- 9 = a (- 3) + b$

$6 = 6 - 2 b + b$

$a = 3 - b$

$24 = a (8) + b$

$27 = a (9) + b$

$- 6 = a (- 2) + b$

$- 9 = a (- 3) + b$

Step 1.1.3.2.2.1.2

Add

$- 2 b$ and

$b$ .

$6 = 6 - b$

$a = 3 - b$

$24 = a (8) + b$

$27 = a (9) + b$

$- 6 = a (- 2) + b$

$- 9 = a (- 3) + b$

$6 = 6 - b$

$a = 3 - b$

$24 = a (8) + b$

$27 = a (9) + b$

$- 6 = a (- 2) + b$

$- 9 = a (- 3) + b$

$6 = 6 - b$

$a = 3 - b$

$24 = a (8) + b$

$27 = a (9) + b$

$- 6 = a (- 2) + b$

$- 9 = a (- 3) + b$

Step 1.1.3.2.3

Replace all occurrences of

$a$ in

$24 = a (8) + b$ with

$3 - b$ .

$24 = (3 - b) (8) + b$

$6 = 6 - b$

$a = 3 - b$

$27 = a (9) + b$

$- 6 = a (- 2) + b$

$- 9 = a (- 3) + b$

Step 1.1.3.2.4

Simplify the right side.

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Step 1.1.3.2.4.1

Simplify

$(3 - b) (8) + b$ .

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Step 1.1.3.2.4.1.1

Simplify each term.

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Step 1.1.3.2.4.1.1.1

Apply the distributive property.

$24 = 3 \cdot 8 - b \cdot 8 + b$

$6 = 6 - b$

$a = 3 - b$

$27 = a (9) + b$

$- 6 = a (- 2) + b$

$- 9 = a (- 3) + b$

Step 1.1.3.2.4.1.1.2

Multiply

$3$ by

$8$ .

$24 = 24 - b \cdot 8 + b$

$6 = 6 - b$

$a = 3 - b$

$27 = a (9) + b$

$- 6 = a (- 2) + b$

$- 9 = a (- 3) + b$

Step 1.1.3.2.4.1.1.3

Multiply

$8$ by

$- 1$ .

$24 = 24 - 8 b + b$

$6 = 6 - b$

$a = 3 - b$

$27 = a (9) + b$

$- 6 = a (- 2) + b$

$- 9 = a (- 3) + b$

$24 = 24 - 8 b + b$

$6 = 6 - b$

$a = 3 - b$

$27 = a (9) + b$

$- 6 = a (- 2) + b$

$- 9 = a (- 3) + b$

Step 1.1.3.2.4.1.2

Add

$- 8 b$ and

$b$ .

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

$27 = a (9) + b$

$- 6 = a (- 2) + b$

$- 9 = a (- 3) + b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

$27 = a (9) + b$

$- 6 = a (- 2) + b$

$- 9 = a (- 3) + b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

$27 = a (9) + b$

$- 6 = a (- 2) + b$

$- 9 = a (- 3) + b$

Step 1.1.3.2.5

Replace all occurrences of

$a$ in

$27 = a (9) + b$ with

$3 - b$ .

$27 = (3 - b) (9) + b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

$- 6 = a (- 2) + b$

$- 9 = a (- 3) + b$

Step 1.1.3.2.6

Simplify the right side.

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Step 1.1.3.2.6.1

Simplify

$(3 - b) (9) + b$ .

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Step 1.1.3.2.6.1.1

Simplify each term.

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Step 1.1.3.2.6.1.1.1

Apply the distributive property.

$27 = 3 \cdot 9 - b \cdot 9 + b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

$- 6 = a (- 2) + b$

$- 9 = a (- 3) + b$

Step 1.1.3.2.6.1.1.2

Multiply

$3$ by

$9$ .

$27 = 27 - b \cdot 9 + b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

$- 6 = a (- 2) + b$

$- 9 = a (- 3) + b$

Step 1.1.3.2.6.1.1.3

Multiply

$9$ by

$- 1$ .

$27 = 27 - 9 b + b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

$- 6 = a (- 2) + b$

$- 9 = a (- 3) + b$

$27 = 27 - 9 b + b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

$- 6 = a (- 2) + b$

$- 9 = a (- 3) + b$

Step 1.1.3.2.6.1.2

Add

$- 9 b$ and

$b$ .

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

$- 6 = a (- 2) + b$

$- 9 = a (- 3) + b$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

$- 6 = a (- 2) + b$

$- 9 = a (- 3) + b$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

$- 6 = a (- 2) + b$

$- 9 = a (- 3) + b$

Step 1.1.3.2.7

Replace all occurrences of

$a$ in

$- 6 = a (- 2) + b$ with

$3 - b$ .

$- 6 = (3 - b) (- 2) + b$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

$- 9 = a (- 3) + b$

Step 1.1.3.2.8

Simplify the right side.

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Step 1.1.3.2.8.1

Simplify

$(3 - b) (- 2) + b$ .

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Step 1.1.3.2.8.1.1

Simplify each term.

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Step 1.1.3.2.8.1.1.1

Apply the distributive property.

$- 6 = 3 \cdot - 2 - b \cdot - 2 + b$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

$- 9 = a (- 3) + b$

Step 1.1.3.2.8.1.1.2

Multiply

$3$ by

$- 2$ .

$- 6 = - 6 - b \cdot - 2 + b$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

$- 9 = a (- 3) + b$

Step 1.1.3.2.8.1.1.3

Multiply

$- 2$ by

$- 1$ .

$- 6 = - 6 + 2 b + b$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

$- 9 = a (- 3) + b$

$- 6 = - 6 + 2 b + b$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

$- 9 = a (- 3) + b$

Step 1.1.3.2.8.1.2

Add

$2 b$ and

$b$ .

$- 6 = - 6 + 3 b$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

$- 9 = a (- 3) + b$

$- 6 = - 6 + 3 b$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

$- 9 = a (- 3) + b$

$- 6 = - 6 + 3 b$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

$- 9 = a (- 3) + b$

Step 1.1.3.2.9

Replace all occurrences of

$a$ in

$- 9 = a (- 3) + b$ with

$3 - b$ .

$- 9 = (3 - b) (- 3) + b$

$- 6 = - 6 + 3 b$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

Step 1.1.3.2.10

Simplify the right side.

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Step 1.1.3.2.10.1

Simplify

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

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Step 1.1.3.2.10.1.1

Simplify each term.

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Step 1.1.3.2.10.1.1.1

Apply the distributive property.

$- 9 = 3 \cdot - 3 - b \cdot - 3 + b$

$- 6 = - 6 + 3 b$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

Step 1.1.3.2.10.1.1.2

Multiply

$3$ by

$- 3$ .

$- 9 = - 9 - b \cdot - 3 + b$

$- 6 = - 6 + 3 b$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

Step 1.1.3.2.10.1.1.3

Multiply

$- 3$ by

$- 1$ .

$- 9 = - 9 + 3 b + b$

$- 6 = - 6 + 3 b$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

$- 9 = - 9 + 3 b + b$

$- 6 = - 6 + 3 b$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

Step 1.1.3.2.10.1.2

Add

$3 b$ and

$b$ .

$- 9 = - 9 + 4 b$

$- 6 = - 6 + 3 b$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

$- 9 = - 9 + 4 b$

$- 6 = - 6 + 3 b$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

$- 9 = - 9 + 4 b$

$- 6 = - 6 + 3 b$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

$- 9 = - 9 + 4 b$

$- 6 = - 6 + 3 b$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

Step 1.1.3.3

Solve for

$b$ in

$- 9 = - 9 + 4 b$ .

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Step 1.1.3.3.1

Rewrite the equation as

$- 9 + 4 b = - 9$ .

$- 9 + 4 b = - 9$

$- 6 = - 6 + 3 b$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

Step 1.1.3.3.2

Move all terms not containing

$b$ to the right side of the equation.

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Step 1.1.3.3.2.1

Add

$9$ to both sides of the equation.

$4 b = - 9 + 9$

$- 6 = - 6 + 3 b$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

Step 1.1.3.3.2.2

Add

$- 9$ and

$9$ .

$4 b = 0$

$- 6 = - 6 + 3 b$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

$4 b = 0$

$- 6 = - 6 + 3 b$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

Step 1.1.3.3.3

Divide each term in

$4 b = 0$ by

$4$ and simplify.

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Step 1.1.3.3.3.1

Divide each term in

$4 b = 0$ by

$4$ .

$\frac{4 b}{4} = \frac{0}{4}$

$- 6 = - 6 + 3 b$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

Step 1.1.3.3.3.2

Simplify the left side.

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Step 1.1.3.3.3.2.1

Cancel the common factor of

$4$ .

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Step 1.1.3.3.3.2.1.1

Cancel the common factor.

$\frac{4 b}{4} = \frac{0}{4}$

$- 6 = - 6 + 3 b$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

Step 1.1.3.3.3.2.1.2

Divide

$b$ by

$1$ .

$b = \frac{0}{4}$

$- 6 = - 6 + 3 b$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

$b = \frac{0}{4}$

$- 6 = - 6 + 3 b$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

$b = \frac{0}{4}$

$- 6 = - 6 + 3 b$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

Step 1.1.3.3.3.3

Simplify the right side.

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Step 1.1.3.3.3.3.1

Divide

$0$ by

$4$ .

$b = 0$

$- 6 = - 6 + 3 b$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

$b = 0$

$- 6 = - 6 + 3 b$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

$b = 0$

$- 6 = - 6 + 3 b$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

$b = 0$

$- 6 = - 6 + 3 b$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

Step 1.1.3.4

Replace all occurrences of

$b$ with

$0$ in each equation.

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Step 1.1.3.4.1

Replace all occurrences of

$b$ in

$- 6 = - 6 + 3 b$ with

$0$ .

$- 6 = - 6 + 3 (0)$

$b = 0$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

Step 1.1.3.4.2

Simplify the right side.

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Step 1.1.3.4.2.1

Simplify

$- 6 + 3 (0)$ .

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Step 1.1.3.4.2.1.1

Multiply

$3$ by

$0$ .

$- 6 = - 6 + 0$

$b = 0$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

Step 1.1.3.4.2.1.2

Add

$- 6$ and

$0$ .

$- 6 = - 6$

$b = 0$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

$- 6 = - 6$

$b = 0$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

$- 6 = - 6$

$b = 0$

$27 = 27 - 8 b$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

Step 1.1.3.4.3

Replace all occurrences of

$b$ in

$27 = 27 - 8 b$ with

$0$ .

$27 = 27 - 8 \cdot 0$

$- 6 = - 6$

$b = 0$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

Step 1.1.3.4.4

Simplify the right side.

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Step 1.1.3.4.4.1

Simplify

$27 - 8 \cdot 0$ .

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Step 1.1.3.4.4.1.1

Multiply

$- 8$ by

$0$ .

$27 = 27 + 0$

$- 6 = - 6$

$b = 0$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

Step 1.1.3.4.4.1.2

Add

$27$ and

$0$ .

$27 = 27$

$- 6 = - 6$

$b = 0$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

$27 = 27$

$- 6 = - 6$

$b = 0$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

$27 = 27$

$- 6 = - 6$

$b = 0$

$24 = 24 - 7 b$

$6 = 6 - b$

$a = 3 - b$

Step 1.1.3.4.5

Replace all occurrences of

$b$ in

$24 = 24 - 7 b$ with

$0$ .

$24 = 24 - 7 \cdot 0$

$27 = 27$

$- 6 = - 6$

$b = 0$

$6 = 6 - b$

$a = 3 - b$

Step 1.1.3.4.6

Simplify the right side.

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Step 1.1.3.4.6.1

Simplify

$24 - 7 \cdot 0$ .

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Step 1.1.3.4.6.1.1

Multiply

$- 7$ by

$0$ .

$24 = 24 + 0$

$27 = 27$

$- 6 = - 6$

$b = 0$

$6 = 6 - b$

$a = 3 - b$

Step 1.1.3.4.6.1.2

Add

$24$ and

$0$ .

$24 = 24$

$27 = 27$

$- 6 = - 6$

$b = 0$

$6 = 6 - b$

$a = 3 - b$

$24 = 24$

$27 = 27$

$- 6 = - 6$

$b = 0$

$6 = 6 - b$

$a = 3 - b$

$24 = 24$

$27 = 27$

$- 6 = - 6$

$b = 0$

$6 = 6 - b$

$a = 3 - b$

Step 1.1.3.4.7

Replace all occurrences of

$b$ in

$6 = 6 - b$ with

$0$ .

$6 = 6 - (0)$

$24 = 24$

$27 = 27$

$- 6 = - 6$

$b = 0$

$a = 3 - b$

Step 1.1.3.4.8

Simplify the right side.

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Step 1.1.3.4.8.1

Subtract

$0$ from

$6$ .

$6 = 6$

$24 = 24$

$27 = 27$

$- 6 = - 6$

$b = 0$

$a = 3 - b$

$6 = 6$

$24 = 24$

$27 = 27$

$- 6 = - 6$

$b = 0$

$a = 3 - b$

Step 1.1.3.4.9

Replace all occurrences of

$b$ in

$a = 3 - b$ with

$0$ .

$a = 3 - (0)$

$6 = 6$

$24 = 24$

$27 = 27$

$- 6 = - 6$

$b = 0$

Step 1.1.3.4.10

Simplify the right side.

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Step 1.1.3.4.10.1

Subtract

$0$ from

$3$ .

$a = 3$

$6 = 6$

$24 = 24$

$27 = 27$

$- 6 = - 6$

$b = 0$

$a = 3$

$6 = 6$

$24 = 24$

$27 = 27$

$- 6 = - 6$

$b = 0$

$a = 3$

$6 = 6$

$24 = 24$

$27 = 27$

$- 6 = - 6$

$b = 0$

Step 1.1.3.5

Remove any equations from the system that are always true.

$a = 3$

$b = 0$

Step 1.1.3.6

List all of the solutions.

$a = 3, b = 0$

Step 1.1.4

Calculate the value of

$y$ using each

$x$ value in the relation and compare this value to the given

$y$ value in the relation.

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Step 1.1.4.1

Calculate the value of

$y$ when

$a = 3$ ,

$b = 0$ , and

$x = 2$ .

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Step 1.1.4.1.1

Multiply

$3$ by

$2$ .

$y = 6 + 0$

Step 1.1.4.1.2

Add

$6$ and

$0$ .

$y = 6$

Step 1.1.4.2

If the table has a linear function rule,

$y = y$ for the corresponding

$x$ value,

$x = 2$ . This check passes since

$y = 6$ and

$y = 6$ .

$6 = 6$

Step 1.1.4.3

Calculate the value of

$y$ when

$a = 3$ ,

$b = 0$ , and

$x = 8$ .

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Step 1.1.4.3.1

Multiply

$3$ by

$8$ .

$y = 24 + 0$

Step 1.1.4.3.2

Add

$24$ and

$0$ .

$y = 24$

Step 1.1.4.4

If the table has a linear function rule,

$y = y$ for the corresponding

$x$ value,

$x = 8$ . This check passes since

$y = 24$ and

$y = 24$ .

$24 = 24$

Step 1.1.4.5

Calculate the value of

$y$ when

$a = 3$ ,

$b = 0$ , and

$x = 9$ .

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Step 1.1.4.5.1

Multiply

$3$ by

$9$ .

$y = 27 + 0$

Step 1.1.4.5.2

Add

$27$ and

$0$ .

$y = 27$

Step 1.1.4.6

If the table has a linear function rule,

$y = y$ for the corresponding

$x$ value,

$x = 9$ . This check passes since

$y = 27$ and

$y = 27$ .

$27 = 27$

Step 1.1.4.7

Calculate the value of

$y$ when

$a = 3$ ,

$b = 0$ , and

$x = - 2$ .

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Step 1.1.4.7.1

Multiply

$3$ by

$- 2$ .

$y = - 6 + 0$

Step 1.1.4.7.2

Add

$- 6$ and

$0$ .

$y = - 6$

Step 1.1.4.8

If the table has a linear function rule,

$y = y$ for the corresponding

$x$ value,

$x = - 2$ . This check passes since

$y = - 6$ and

$y = - 6$ .

$- 6 = - 6$

Step 1.1.4.9

Calculate the value of

$y$ when

$a = 3$ ,

$b = 0$ , and

$x = - 3$ .

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Step 1.1.4.9.1

Multiply

$3$ by

$- 3$ .

$y = - 9 + 0$

Step 1.1.4.9.2

Add

$- 9$ and

$0$ .

$y = - 9$

Step 1.1.4.10

If the table has a linear function rule,

$y = y$ for the corresponding

$x$ value,

$x = - 3$ . This check passes since

$y = - 9$ and

$y = - 9$ .

$- 9 = - 9$

Step 1.1.4.11

Calculate the value of

$y$ when

$a = 3$ ,

$b = 0$ , and

$x = 1$ .

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Step 1.1.4.11.1

Multiply

$3$ by

$1$ .

$y = 3 + 0$

Step 1.1.4.11.2

Add

$3$ and

$0$ .

$y = 3$

Step 1.1.4.12

If the table has a linear function rule,

$y = y$ for the corresponding

$x$ value,

$x = 1$ . This check passes since

$y = 3$ and

$y = 3$ .

$3 = 3$

Step 1.1.4.13

Calculate the value of

$y$ when

$a = 3$ ,

$b = 0$ , and

$x = 1$ .

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Step 1.1.4.13.1

Multiply

$3$ by

$1$ .

$y = 3 + 0$

Step 1.1.4.13.2

Add

$3$ and

$0$ .

$y = 3$

Step 1.1.4.14

If the table has a linear function rule,

$y = y$ for the corresponding

$x$ value,

$x = 1$ . This check passes since

$y = 3$ and

$y = 3$ .

$3 = 3$

Step 1.1.4.15

Calculate the value of

$y$ when

$a = 3$ ,

$b = 0$ , and

$x = 1$ .

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Step 1.1.4.15.1

Multiply

$3$ by

$1$ .

$y = 3 + 0$

Step 1.1.4.15.2

Add

$3$ and

$0$ .

$y = 3$

Step 1.1.4.16

If the table has a linear function rule,

$y = y$ for the corresponding

$x$ value,

$x = 1$ . This check passes since

$y = 3$ and

$y = 3$ .

$3 = 3$

Step 1.1.4.17

Since

$y = y$ for the corresponding

$x$ values, the function is linear.

The function is linear

Step 1.2

Since all

$y = y$ , the function is linear and follows the form

$y = 3 x$ .

$y = 3 x$

Step 2

Find

$x$ .

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Step 2.1

Use the function rule equation to find

$x$ .

$x = 3 (1)$

Step 2.2

Simplify.

$x = 3$

Step 3

Find

$z$ .

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Step 3.1

Use the function rule equation to find

$z$ .

$z = 3 (7)$

Step 3.2

Simplify.

$z = 21$

Step 4

List all of the solutions.

$x = 3 z = 21$

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