Examples

$\begin{array}{c} x & q (x) \\ 1 & 2 \\ 2 & 3 \\ 3 & 4 \\ 4 & 5 \\ 5 & 6 \\ 6 & 7 \\ 7 & 8 \end{array}$

Step 1

Check if the function rule is linear.

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Step 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 1.2

Build a set of equations from the table such that $q (x) = a x + b$ .

$2 = a (1) + b 3 = a (2) + b 4 = a (3) + b 5 = a (4) + b 6 = a (5) + b 7 = a (6) + b 8 = a (7) + b$

Step 1.3

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

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Step 1.3.1

Solve for $a$ in $2 = a + b$ .

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Step 1.3.1.1

Rewrite the equation as $a + b = 2$ .

$a + b = 2$

$3 = a (2) + b$

$4 = a (3) + b$

$5 = a (4) + b$

$6 = a (5) + b$

$7 = a (6) + b$

$8 = a (7) + b$

Step 1.3.1.2

Subtract $b$ from both sides of the equation.

$a = 2 - b$

$3 = a (2) + b$

$4 = a (3) + b$

$5 = a (4) + b$

$6 = a (5) + b$

$7 = a (6) + b$

$8 = a (7) + b$

$a = 2 - b$

$3 = a (2) + b$

$4 = a (3) + b$

$5 = a (4) + b$

$6 = a (5) + b$

$7 = a (6) + b$

$8 = a (7) + b$

Step 1.3.2

Replace all occurrences of $a$ with $2 - b$ in each equation.

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Step 1.3.2.1

Replace all occurrences of $a$ in $3 = a (2) + b$ with $2 - b$ .

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

$a = 2 - b$

$4 = a (3) + b$

$5 = a (4) + b$

$6 = a (5) + b$

$7 = a (6) + b$

$8 = a (7) + b$

Step 1.3.2.2

Simplify the right side.

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Step 1.3.2.2.1

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

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Step 1.3.2.2.1.1

Simplify each term.

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Step 1.3.2.2.1.1.1

Apply the distributive property.

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

$a = 2 - b$

$4 = a (3) + b$

$5 = a (4) + b$

$6 = a (5) + b$

$7 = a (6) + b$

$8 = a (7) + b$

Step 1.3.2.2.1.1.2

Multiply $2$ by $2$ .

$3 = 4 - b \cdot 2 + b$

$a = 2 - b$

$4 = a (3) + b$

$5 = a (4) + b$

$6 = a (5) + b$

$7 = a (6) + b$

$8 = a (7) + b$

Step 1.3.2.2.1.1.3

Multiply $2$ by $- 1$ .

$3 = 4 - 2 b + b$

$a = 2 - b$

$4 = a (3) + b$

$5 = a (4) + b$

$6 = a (5) + b$

$7 = a (6) + b$

$8 = a (7) + b$

$3 = 4 - 2 b + b$

$a = 2 - b$

$4 = a (3) + b$

$5 = a (4) + b$

$6 = a (5) + b$

$7 = a (6) + b$

$8 = a (7) + b$

Step 1.3.2.2.1.2

Add $- 2 b$ and $b$ .

$3 = 4 - b$

$a = 2 - b$

$4 = a (3) + b$

$5 = a (4) + b$

$6 = a (5) + b$

$7 = a (6) + b$

$8 = a (7) + b$

$3 = 4 - b$

$a = 2 - b$

$4 = a (3) + b$

$5 = a (4) + b$

$6 = a (5) + b$

$7 = a (6) + b$

$8 = a (7) + b$

$3 = 4 - b$

$a = 2 - b$

$4 = a (3) + b$

$5 = a (4) + b$

$6 = a (5) + b$

$7 = a (6) + b$

$8 = a (7) + b$

Step 1.3.2.3

Replace all occurrences of $a$ in $4 = a (3) + b$ with $2 - b$ .

$4 = (2 - b) (3) + b$

$3 = 4 - b$

$a = 2 - b$

$5 = a (4) + b$

$6 = a (5) + b$

$7 = a (6) + b$

$8 = a (7) + b$

Step 1.3.2.4

Simplify the right side.

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Step 1.3.2.4.1

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

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Step 1.3.2.4.1.1

Simplify each term.

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Step 1.3.2.4.1.1.1

Apply the distributive property.

$4 = 2 \cdot 3 - b \cdot 3 + b$

$3 = 4 - b$

$a = 2 - b$

$5 = a (4) + b$

$6 = a (5) + b$

$7 = a (6) + b$

$8 = a (7) + b$

Step 1.3.2.4.1.1.2

Multiply $2$ by $3$ .

$4 = 6 - b \cdot 3 + b$

$3 = 4 - b$

$a = 2 - b$

$5 = a (4) + b$

$6 = a (5) + b$

$7 = a (6) + b$

$8 = a (7) + b$

Step 1.3.2.4.1.1.3

Multiply $3$ by $- 1$ .

$4 = 6 - 3 b + b$

$3 = 4 - b$

$a = 2 - b$

$5 = a (4) + b$

$6 = a (5) + b$

$7 = a (6) + b$

$8 = a (7) + b$

$4 = 6 - 3 b + b$

$3 = 4 - b$

$a = 2 - b$

$5 = a (4) + b$

$6 = a (5) + b$

$7 = a (6) + b$

$8 = a (7) + b$

Step 1.3.2.4.1.2

Add $- 3 b$ and $b$ .

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$5 = a (4) + b$

$6 = a (5) + b$

$7 = a (6) + b$

$8 = a (7) + b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$5 = a (4) + b$

$6 = a (5) + b$

$7 = a (6) + b$

$8 = a (7) + b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$5 = a (4) + b$

$6 = a (5) + b$

$7 = a (6) + b$

$8 = a (7) + b$

Step 1.3.2.5

Replace all occurrences of $a$ in $5 = a (4) + b$ with $2 - b$ .

$5 = (2 - b) (4) + b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$6 = a (5) + b$

$7 = a (6) + b$

$8 = a (7) + b$

Step 1.3.2.6

Simplify the right side.

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Step 1.3.2.6.1

Simplify $(2 - b) (4) + b$ .

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Step 1.3.2.6.1.1

Simplify each term.

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Step 1.3.2.6.1.1.1

Apply the distributive property.

$5 = 2 \cdot 4 - b \cdot 4 + b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$6 = a (5) + b$

$7 = a (6) + b$

$8 = a (7) + b$

Step 1.3.2.6.1.1.2

Multiply $2$ by $4$ .

$5 = 8 - b \cdot 4 + b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$6 = a (5) + b$

$7 = a (6) + b$

$8 = a (7) + b$

Step 1.3.2.6.1.1.3

Multiply $4$ by $- 1$ .

$5 = 8 - 4 b + b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$6 = a (5) + b$

$7 = a (6) + b$

$8 = a (7) + b$

$5 = 8 - 4 b + b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$6 = a (5) + b$

$7 = a (6) + b$

$8 = a (7) + b$

Step 1.3.2.6.1.2

Add $- 4 b$ and $b$ .

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$6 = a (5) + b$

$7 = a (6) + b$

$8 = a (7) + b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$6 = a (5) + b$

$7 = a (6) + b$

$8 = a (7) + b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$6 = a (5) + b$

$7 = a (6) + b$

$8 = a (7) + b$

Step 1.3.2.7

Replace all occurrences of $a$ in $6 = a (5) + b$ with $2 - b$ .

$6 = (2 - b) (5) + b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$7 = a (6) + b$

$8 = a (7) + b$

Step 1.3.2.8

Simplify the right side.

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Step 1.3.2.8.1

Simplify $(2 - b) (5) + b$ .

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Step 1.3.2.8.1.1

Simplify each term.

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Step 1.3.2.8.1.1.1

Apply the distributive property.

$6 = 2 \cdot 5 - b \cdot 5 + b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$7 = a (6) + b$

$8 = a (7) + b$

Step 1.3.2.8.1.1.2

Multiply $2$ by $5$ .

$6 = 10 - b \cdot 5 + b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$7 = a (6) + b$

$8 = a (7) + b$

Step 1.3.2.8.1.1.3

Multiply $5$ by $- 1$ .

$6 = 10 - 5 b + b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$7 = a (6) + b$

$8 = a (7) + b$

$6 = 10 - 5 b + b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$7 = a (6) + b$

$8 = a (7) + b$

Step 1.3.2.8.1.2

Add $- 5 b$ and $b$ .

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$7 = a (6) + b$

$8 = a (7) + b$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$7 = a (6) + b$

$8 = a (7) + b$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$7 = a (6) + b$

$8 = a (7) + b$

Step 1.3.2.9

Replace all occurrences of $a$ in $7 = a (6) + b$ with $2 - b$ .

$7 = (2 - b) (6) + b$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$8 = a (7) + b$

Step 1.3.2.10

Simplify the right side.

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Step 1.3.2.10.1

Simplify $(2 - b) (6) + b$ .

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Step 1.3.2.10.1.1

Simplify each term.

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Step 1.3.2.10.1.1.1

Apply the distributive property.

$7 = 2 \cdot 6 - b \cdot 6 + b$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$8 = a (7) + b$

Step 1.3.2.10.1.1.2

Multiply $2$ by $6$ .

$7 = 12 - b \cdot 6 + b$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$8 = a (7) + b$

Step 1.3.2.10.1.1.3

Multiply $6$ by $- 1$ .

$7 = 12 - 6 b + b$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$8 = a (7) + b$

$7 = 12 - 6 b + b$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$8 = a (7) + b$

Step 1.3.2.10.1.2

Add $- 6 b$ and $b$ .

$7 = 12 - 5 b$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$8 = a (7) + b$

$7 = 12 - 5 b$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$8 = a (7) + b$

$7 = 12 - 5 b$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$8 = a (7) + b$

Step 1.3.2.11

Replace all occurrences of $a$ in $8 = a (7) + b$ with $2 - b$ .

$8 = (2 - b) (7) + b$

$7 = 12 - 5 b$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

Step 1.3.2.12

Simplify the right side.

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Step 1.3.2.12.1

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

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Step 1.3.2.12.1.1

Simplify each term.

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Step 1.3.2.12.1.1.1

Apply the distributive property.

$8 = 2 \cdot 7 - b \cdot 7 + b$

$7 = 12 - 5 b$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

Step 1.3.2.12.1.1.2

Multiply $2$ by $7$ .

$8 = 14 - b \cdot 7 + b$

$7 = 12 - 5 b$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

Step 1.3.2.12.1.1.3

Multiply $7$ by $- 1$ .

$8 = 14 - 7 b + b$

$7 = 12 - 5 b$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$8 = 14 - 7 b + b$

$7 = 12 - 5 b$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

Step 1.3.2.12.1.2

Add $- 7 b$ and $b$ .

$8 = 14 - 6 b$

$7 = 12 - 5 b$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$8 = 14 - 6 b$

$7 = 12 - 5 b$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$8 = 14 - 6 b$

$7 = 12 - 5 b$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$8 = 14 - 6 b$

$7 = 12 - 5 b$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

Step 1.3.3

Solve for $b$ in $8 = 14 - 6 b$ .

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Step 1.3.3.1

Rewrite the equation as $14 - 6 b = 8$ .

$14 - 6 b = 8$

$7 = 12 - 5 b$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

Step 1.3.3.2

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

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Step 1.3.3.2.1

Subtract $14$ from both sides of the equation.

$- 6 b = 8 - 14$

$7 = 12 - 5 b$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

Step 1.3.3.2.2

Subtract $14$ from $8$ .

$- 6 b = - 6$

$7 = 12 - 5 b$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$- 6 b = - 6$

$7 = 12 - 5 b$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

Step 1.3.3.3

Divide each term in $- 6 b = - 6$ by $- 6$ and simplify.

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Step 1.3.3.3.1

Divide each term in $- 6 b = - 6$ by $- 6$ .

$\frac{- 6 b}{- 6} = \frac{- 6}{- 6}$

$7 = 12 - 5 b$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

Step 1.3.3.3.2

Simplify the left side.

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Step 1.3.3.3.2.1

Cancel the common factor of $- 6$ .

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Step 1.3.3.3.2.1.1

Cancel the common factor.

$\frac{- 6 b}{- 6} = \frac{- 6}{- 6}$

$7 = 12 - 5 b$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

Step 1.3.3.3.2.1.2

Divide $b$ by $1$ .

$b = \frac{- 6}{- 6}$

$7 = 12 - 5 b$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$b = \frac{- 6}{- 6}$

$7 = 12 - 5 b$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$b = \frac{- 6}{- 6}$

$7 = 12 - 5 b$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

Step 1.3.3.3.3

Simplify the right side.

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Step 1.3.3.3.3.1

Divide $- 6$ by $- 6$ .

$b = 1$

$7 = 12 - 5 b$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$b = 1$

$7 = 12 - 5 b$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$b = 1$

$7 = 12 - 5 b$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$b = 1$

$7 = 12 - 5 b$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

Step 1.3.4

Replace all occurrences of $b$ with $1$ in each equation.

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Step 1.3.4.1

Replace all occurrences of $b$ in $7 = 12 - 5 b$ with $1$ .

$7 = 12 - 5 \cdot 1$

$b = 1$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

Step 1.3.4.2

Simplify the right side.

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Step 1.3.4.2.1

Simplify $12 - 5 \cdot 1$ .

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Step 1.3.4.2.1.1

Multiply $- 5$ by $1$ .

$7 = 12 - 5$

$b = 1$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

Step 1.3.4.2.1.2

Subtract $5$ from $12$ .

$7 = 7$

$b = 1$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$7 = 7$

$b = 1$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$7 = 7$

$b = 1$

$6 = 10 - 4 b$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

Step 1.3.4.3

Replace all occurrences of $b$ in $6 = 10 - 4 b$ with $1$ .

$6 = 10 - 4 \cdot 1$

$7 = 7$

$b = 1$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

Step 1.3.4.4

Simplify the right side.

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Step 1.3.4.4.1

Simplify $10 - 4 \cdot 1$ .

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Step 1.3.4.4.1.1

Multiply $- 4$ by $1$ .

$6 = 10 - 4$

$7 = 7$

$b = 1$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

Step 1.3.4.4.1.2

Subtract $4$ from $10$ .

$6 = 6$

$7 = 7$

$b = 1$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$6 = 6$

$7 = 7$

$b = 1$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$6 = 6$

$7 = 7$

$b = 1$

$5 = 8 - 3 b$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

Step 1.3.4.5

Replace all occurrences of $b$ in $5 = 8 - 3 b$ with $1$ .

$5 = 8 - 3 \cdot 1$

$6 = 6$

$7 = 7$

$b = 1$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

Step 1.3.4.6

Simplify the right side.

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Step 1.3.4.6.1

Simplify $8 - 3 \cdot 1$ .

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Step 1.3.4.6.1.1

Multiply $- 3$ by $1$ .

$5 = 8 - 3$

$6 = 6$

$7 = 7$

$b = 1$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

Step 1.3.4.6.1.2

Subtract $3$ from $8$ .

$5 = 5$

$6 = 6$

$7 = 7$

$b = 1$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$5 = 5$

$6 = 6$

$7 = 7$

$b = 1$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

$5 = 5$

$6 = 6$

$7 = 7$

$b = 1$

$4 = 6 - 2 b$

$3 = 4 - b$

$a = 2 - b$

Step 1.3.4.7

Replace all occurrences of $b$ in $4 = 6 - 2 b$ with $1$ .

$4 = 6 - 2 \cdot 1$

$5 = 5$

$6 = 6$

$7 = 7$

$b = 1$

$3 = 4 - b$

$a = 2 - b$

Step 1.3.4.8

Simplify the right side.

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Step 1.3.4.8.1

Simplify $6 - 2 \cdot 1$ .

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Step 1.3.4.8.1.1

Multiply $- 2$ by $1$ .

$4 = 6 - 2$

$5 = 5$

$6 = 6$

$7 = 7$

$b = 1$

$3 = 4 - b$

$a = 2 - b$

Step 1.3.4.8.1.2

Subtract $2$ from $6$ .

$4 = 4$

$5 = 5$

$6 = 6$

$7 = 7$

$b = 1$

$3 = 4 - b$

$a = 2 - b$

$4 = 4$

$5 = 5$

$6 = 6$

$7 = 7$

$b = 1$

$3 = 4 - b$

$a = 2 - b$

$4 = 4$

$5 = 5$

$6 = 6$

$7 = 7$

$b = 1$

$3 = 4 - b$

$a = 2 - b$

Step 1.3.4.9

Replace all occurrences of $b$ in $3 = 4 - b$ with $1$ .

$3 = 4 - (1)$

$4 = 4$

$5 = 5$

$6 = 6$

$7 = 7$

$b = 1$

$a = 2 - b$

Step 1.3.4.10

Simplify the right side.

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Step 1.3.4.10.1

Simplify $4 - (1)$ .

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Step 1.3.4.10.1.1

Multiply $- 1$ by $1$ .

$3 = 4 - 1$

$4 = 4$

$5 = 5$

$6 = 6$

$7 = 7$

$b = 1$

$a = 2 - b$

Step 1.3.4.10.1.2

Subtract $1$ from $4$ .

$3 = 3$

$4 = 4$

$5 = 5$

$6 = 6$

$7 = 7$

$b = 1$

$a = 2 - b$

$3 = 3$

$4 = 4$

$5 = 5$

$6 = 6$

$7 = 7$

$b = 1$

$a = 2 - b$

$3 = 3$

$4 = 4$

$5 = 5$

$6 = 6$

$7 = 7$

$b = 1$

$a = 2 - b$

Step 1.3.4.11

Replace all occurrences of $b$ in $a = 2 - b$ with $1$ .

$a = 2 - (1)$

$3 = 3$

$4 = 4$

$5 = 5$

$6 = 6$

$7 = 7$

$b = 1$

Step 1.3.4.12

Simplify the right side.

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Step 1.3.4.12.1

Simplify $2 - (1)$ .

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Step 1.3.4.12.1.1

Multiply $- 1$ by $1$ .

$a = 2 - 1$

$3 = 3$

$4 = 4$

$5 = 5$

$6 = 6$

$7 = 7$

$b = 1$

Step 1.3.4.12.1.2

Subtract $1$ from $2$ .

$a = 1$

$3 = 3$

$4 = 4$

$5 = 5$

$6 = 6$

$7 = 7$

$b = 1$

$a = 1$

$3 = 3$

$4 = 4$

$5 = 5$

$6 = 6$

$7 = 7$

$b = 1$

$a = 1$

$3 = 3$

$4 = 4$

$5 = 5$

$6 = 6$

$7 = 7$

$b = 1$

$a = 1$

$3 = 3$

$4 = 4$

$5 = 5$

$6 = 6$

$7 = 7$

$b = 1$

Step 1.3.5

Remove any equations from the system that are always true.

$a = 1$

$b = 1$

Step 1.3.6

List all of the solutions.

$a = 1, b = 1$

Step 1.4

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

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Step 1.4.1

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

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Step 1.4.1.1

Multiply $1$ by $1$ .

$y = 1 + 1$

Step 1.4.1.2

Add $1$ and $1$ .

$y = 2$

Step 1.4.2

If the table has a linear function rule, $y = q (x)$ for the corresponding $x$ value, $x = 1$ . This check passes since $y = 2$ and $q (x) = 2$ .

$2 = 2$

Step 1.4.3

Calculate the value of $y$ when $a = 1$ , $b = 1$ , and $x = 2$ .

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Step 1.4.3.1

Multiply $2$ by $1$ .

$y = 2 + 1$

Step 1.4.3.2

Add $2$ and $1$ .

$y = 3$

Step 1.4.4

If the table has a linear function rule, $y = q (x)$ for the corresponding $x$ value, $x = 2$ . This check passes since $y = 3$ and $q (x) = 3$ .

$3 = 3$

Step 1.4.5

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

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Step 1.4.5.1

Multiply $3$ by $1$ .

$y = 3 + 1$

Step 1.4.5.2

Add $3$ and $1$ .

$y = 4$

Step 1.4.6

If the table has a linear function rule, $y = q (x)$ for the corresponding $x$ value, $x = 3$ . This check passes since $y = 4$ and $q (x) = 4$ .

$4 = 4$

Step 1.4.7

Calculate the value of $y$ when $a = 1$ , $b = 1$ , and $x = 4$ .

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Step 1.4.7.1

Multiply $4$ by $1$ .

$y = 4 + 1$

Step 1.4.7.2

Add $4$ and $1$ .

$y = 5$

Step 1.4.8

If the table has a linear function rule, $y = q (x)$ for the corresponding $x$ value, $x = 4$ . This check passes since $y = 5$ and $q (x) = 5$ .

$5 = 5$

Step 1.4.9

Calculate the value of $y$ when $a = 1$ , $b = 1$ , and $x = 5$ .

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Step 1.4.9.1

Multiply $5$ by $1$ .

$y = 5 + 1$

Step 1.4.9.2

Add $5$ and $1$ .

$y = 6$

Step 1.4.10

If the table has a linear function rule, $y = q (x)$ for the corresponding $x$ value, $x = 5$ . This check passes since $y = 6$ and $q (x) = 6$ .

$6 = 6$

Step 1.4.11

Calculate the value of $y$ when $a = 1$ , $b = 1$ , and $x = 6$ .

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Step 1.4.11.1

Multiply $6$ by $1$ .

$y = 6 + 1$

Step 1.4.11.2

Add $6$ and $1$ .

$y = 7$

Step 1.4.12

If the table has a linear function rule, $y = q (x)$ for the corresponding $x$ value, $x = 6$ . This check passes since $y = 7$ and $q (x) = 7$ .

$7 = 7$

Step 1.4.13

Calculate the value of $y$ when $a = 1$ , $b = 1$ , and $x = 7$ .

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Step 1.4.13.1

Multiply $7$ by $1$ .

$y = 7 + 1$

Step 1.4.13.2

Add $7$ and $1$ .

$y = 8$

Step 1.4.14

If the table has a linear function rule, $y = q (x)$ for the corresponding $x$ value, $x = 7$ . This check passes since $y = 8$ and $q (x) = 8$ .

$8 = 8$

Step 1.4.15

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

The function is linear

Step 2

Since all $y = q (x)$ , the function is linear and follows the form $y = x + 1$ .

$y = x + 1$

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