Examples

$\begin{array}{c} x & q (x) \\ 6 & 12 \\ 3 & 6 \\ 8 & 16 \\ 12 & 24 \end{array}$

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 $q (x) = a x + b$ .

$12 = a (6) + b 6 = a (3) + b 16 = a (8) + b 24 = a (12) + b$

Step 1.3

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

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Step 1.3.1

Solve for $b$ in $12 = a (6) + b$ .

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Step 1.3.1.1

Rewrite the equation as $a (6) + b = 12$ .

$a (6) + b = 12$

$6 = a (3) + b$

$16 = a (8) + b$

$24 = a (12) + b$

Step 1.3.1.2

Move $6$ to the left of $a$ .

$6 a + b = 12$

$6 = a (3) + b$

$16 = a (8) + b$

$24 = a (12) + b$

Step 1.3.1.3

Subtract $6 a$ from both sides of the equation.

$b = 12 - 6 a$

$6 = a (3) + b$

$16 = a (8) + b$

$24 = a (12) + b$

$b = 12 - 6 a$

$6 = a (3) + b$

$16 = a (8) + b$

$24 = a (12) + b$

Step 1.3.2

Replace all occurrences of $b$ with $12 - 6 a$ in each equation.

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Step 1.3.2.1

Replace all occurrences of $b$ in $6 = a (3) + b$ with $12 - 6 a$ .

$6 = a (3) + 12 - 6 a$

$b = 12 - 6 a$

$16 = a (8) + b$

$24 = a (12) + b$

Step 1.3.2.2

Simplify $6 = a (3) + 12 - 6 a$ .

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Step 1.3.2.2.1

Simplify the left side.

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Step 1.3.2.2.1.1

Remove parentheses.

$6 = a (3) + 12 - 6 a$

$b = 12 - 6 a$

$16 = a (8) + b$

$24 = a (12) + b$

$6 = a (3) + 12 - 6 a$

$b = 12 - 6 a$

$16 = a (8) + b$

$24 = a (12) + b$

Step 1.3.2.2.2

Simplify the right side.

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Step 1.3.2.2.2.1

Simplify $a (3) + 12 - 6 a$ .

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Step 1.3.2.2.2.1.1

Move $3$ to the left of $a$ .

$6 = 3 a + 12 - 6 a$

$b = 12 - 6 a$

$16 = a (8) + b$

$24 = a (12) + b$

Step 1.3.2.2.2.1.2

Subtract $6 a$ from $3 a$ .

$6 = - 3 a + 12$

$b = 12 - 6 a$

$16 = a (8) + b$

$24 = a (12) + b$

$6 = - 3 a + 12$

$b = 12 - 6 a$

$16 = a (8) + b$

$24 = a (12) + b$

$6 = - 3 a + 12$

$b = 12 - 6 a$

$16 = a (8) + b$

$24 = a (12) + b$

$6 = - 3 a + 12$

$b = 12 - 6 a$

$16 = a (8) + b$

$24 = a (12) + b$

Step 1.3.2.3

Replace all occurrences of $b$ in $16 = a (8) + b$ with $12 - 6 a$ .

$16 = a (8) + 12 - 6 a$

$6 = - 3 a + 12$

$b = 12 - 6 a$

$24 = a (12) + b$

Step 1.3.2.4

Simplify $16 = a (8) + 12 - 6 a$ .

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Step 1.3.2.4.1

Simplify the left side.

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Step 1.3.2.4.1.1

Remove parentheses.

$16 = a (8) + 12 - 6 a$

$6 = - 3 a + 12$

$b = 12 - 6 a$

$24 = a (12) + b$

$16 = a (8) + 12 - 6 a$

$6 = - 3 a + 12$

$b = 12 - 6 a$

$24 = a (12) + b$

Step 1.3.2.4.2

Simplify the right side.

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Step 1.3.2.4.2.1

Simplify $a (8) + 12 - 6 a$ .

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Step 1.3.2.4.2.1.1

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

$16 = 8 a + 12 - 6 a$

$6 = - 3 a + 12$

$b = 12 - 6 a$

$24 = a (12) + b$

Step 1.3.2.4.2.1.2

Subtract $6 a$ from $8 a$ .

$16 = 2 a + 12$

$6 = - 3 a + 12$

$b = 12 - 6 a$

$24 = a (12) + b$

$16 = 2 a + 12$

$6 = - 3 a + 12$

$b = 12 - 6 a$

$24 = a (12) + b$

$16 = 2 a + 12$

$6 = - 3 a + 12$

$b = 12 - 6 a$

$24 = a (12) + b$

$16 = 2 a + 12$

$6 = - 3 a + 12$

$b = 12 - 6 a$

$24 = a (12) + b$

Step 1.3.2.5

Replace all occurrences of $b$ in $24 = a (12) + b$ with $12 - 6 a$ .

$24 = a (12) + 12 - 6 a$

$16 = 2 a + 12$

$6 = - 3 a + 12$

$b = 12 - 6 a$

Step 1.3.2.6

Simplify $24 = a (12) + 12 - 6 a$ .

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Step 1.3.2.6.1

Simplify the left side.

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Step 1.3.2.6.1.1

Remove parentheses.

$24 = a (12) + 12 - 6 a$

$16 = 2 a + 12$

$6 = - 3 a + 12$

$b = 12 - 6 a$

$24 = a (12) + 12 - 6 a$

$16 = 2 a + 12$

$6 = - 3 a + 12$

$b = 12 - 6 a$

Step 1.3.2.6.2

Simplify the right side.

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Step 1.3.2.6.2.1

Simplify $a (12) + 12 - 6 a$ .

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Step 1.3.2.6.2.1.1

Move $12$ to the left of $a$ .

$24 = 12 a + 12 - 6 a$

$16 = 2 a + 12$

$6 = - 3 a + 12$

$b = 12 - 6 a$

Step 1.3.2.6.2.1.2

Subtract $6 a$ from $12 a$ .

$24 = 6 a + 12$

$16 = 2 a + 12$

$6 = - 3 a + 12$

$b = 12 - 6 a$

$24 = 6 a + 12$

$16 = 2 a + 12$

$6 = - 3 a + 12$

$b = 12 - 6 a$

$24 = 6 a + 12$

$16 = 2 a + 12$

$6 = - 3 a + 12$

$b = 12 - 6 a$

$24 = 6 a + 12$

$16 = 2 a + 12$

$6 = - 3 a + 12$

$b = 12 - 6 a$

$24 = 6 a + 12$

$16 = 2 a + 12$

$6 = - 3 a + 12$

$b = 12 - 6 a$

Step 1.3.3

Solve for $a$ in $24 = 6 a + 12$ .

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Step 1.3.3.1

Rewrite the equation as $6 a + 12 = 24$ .

$6 a + 12 = 24$

$16 = 2 a + 12$

$6 = - 3 a + 12$

$b = 12 - 6 a$

Step 1.3.3.2

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

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Step 1.3.3.2.1

Subtract $12$ from both sides of the equation.

$6 a = 24 - 12$

$16 = 2 a + 12$

$6 = - 3 a + 12$

$b = 12 - 6 a$

Step 1.3.3.2.2

Subtract $12$ from $24$ .

$6 a = 12$

$16 = 2 a + 12$

$6 = - 3 a + 12$

$b = 12 - 6 a$

$6 a = 12$

$16 = 2 a + 12$

$6 = - 3 a + 12$

$b = 12 - 6 a$

Step 1.3.3.3

Divide each term in $6 a = 12$ by $6$ and simplify.

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Step 1.3.3.3.1

Divide each term in $6 a = 12$ by $6$ .

$\frac{6 a}{6} = \frac{12}{6}$

$16 = 2 a + 12$

$6 = - 3 a + 12$

$b = 12 - 6 a$

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.

$\frac{6 a}{6} = \frac{12}{6}$

$16 = 2 a + 12$

$6 = - 3 a + 12$

$b = 12 - 6 a$

Step 1.3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{12}{6}$

$16 = 2 a + 12$

$6 = - 3 a + 12$

$b = 12 - 6 a$

$a = \frac{12}{6}$

$16 = 2 a + 12$

$6 = - 3 a + 12$

$b = 12 - 6 a$

$a = \frac{12}{6}$

$16 = 2 a + 12$

$6 = - 3 a + 12$

$b = 12 - 6 a$

Step 1.3.3.3.3

Simplify the right side.

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Step 1.3.3.3.3.1

Divide $12$ by $6$ .

$a = 2$

$16 = 2 a + 12$

$6 = - 3 a + 12$

$b = 12 - 6 a$

$a = 2$

$16 = 2 a + 12$

$6 = - 3 a + 12$

$b = 12 - 6 a$

$a = 2$

$16 = 2 a + 12$

$6 = - 3 a + 12$

$b = 12 - 6 a$

$a = 2$

$16 = 2 a + 12$

$6 = - 3 a + 12$

$b = 12 - 6 a$

Step 1.3.4

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

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Step 1.3.4.1

Replace all occurrences of $a$ in $16 = 2 a + 12$ with $2$ .

$16 = 2 (2) + 12$

$a = 2$

$6 = - 3 a + 12$

$b = 12 - 6 a$

Step 1.3.4.2

Simplify the right side.

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Step 1.3.4.2.1

Simplify $2 (2) + 12$ .

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Step 1.3.4.2.1.1

Multiply $2$ by $2$ .

$16 = 4 + 12$

$a = 2$

$6 = - 3 a + 12$

$b = 12 - 6 a$

Step 1.3.4.2.1.2

Add $4$ and $12$ .

$16 = 16$

$a = 2$

$6 = - 3 a + 12$

$b = 12 - 6 a$

$16 = 16$

$a = 2$

$6 = - 3 a + 12$

$b = 12 - 6 a$

$16 = 16$

$a = 2$

$6 = - 3 a + 12$

$b = 12 - 6 a$

Step 1.3.4.3

Replace all occurrences of $a$ in $6 = - 3 a + 12$ with $2$ .

$6 = - 3 \cdot 2 + 12$

$16 = 16$

$a = 2$

$b = 12 - 6 a$

Step 1.3.4.4

Simplify the right side.

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Step 1.3.4.4.1

Simplify $- 3 \cdot 2 + 12$ .

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Step 1.3.4.4.1.1

Multiply $- 3$ by $2$ .

$6 = - 6 + 12$

$16 = 16$

$a = 2$

$b = 12 - 6 a$

Step 1.3.4.4.1.2

Add $- 6$ and $12$ .

$6 = 6$

$16 = 16$

$a = 2$

$b = 12 - 6 a$

$6 = 6$

$16 = 16$

$a = 2$

$b = 12 - 6 a$

$6 = 6$

$16 = 16$

$a = 2$

$b = 12 - 6 a$

Step 1.3.4.5

Replace all occurrences of $a$ in $b = 12 - 6 a$ with $2$ .

$b = 12 - 6 \cdot 2$

$6 = 6$

$16 = 16$

$a = 2$

Step 1.3.4.6

Simplify the right side.

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Step 1.3.4.6.1

Simplify $12 - 6 \cdot 2$ .

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Step 1.3.4.6.1.1

Multiply $- 6$ by $2$ .

$b = 12 - 12$

$6 = 6$

$16 = 16$

$a = 2$

Step 1.3.4.6.1.2

Subtract $12$ from $12$ .

$b = 0$

$6 = 6$

$16 = 16$

$a = 2$

$b = 0$

$6 = 6$

$16 = 16$

$a = 2$

$b = 0$

$6 = 6$

$16 = 16$

$a = 2$

$b = 0$

$6 = 6$

$16 = 16$

$a = 2$

Step 1.3.5

Remove any equations from the system that are always true.

$b = 0$

$a = 2$

Step 1.3.6

List all of the solutions.

$b = 0, a = 2$

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 $a = 2$ , $b = 0$ , and $x = 6$ .

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Step 1.4.1.1

Multiply $2$ by $6$ .

$y = 12 + 0$

Step 1.4.1.2

Add $12$ and $0$ .

$y = 12$

Step 1.4.2

If the table has a linear function rule, $y = q (x)$ for the corresponding $x$ value, $x = 6$ . This check passes since $y = 12$ and $q (x) = 12$ .

$12 = 12$

Step 1.4.3

Calculate the value of $y$ when $a = 2$ , $b = 0$ , and $x = 3$ .

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Step 1.4.3.1

Multiply $2$ by $3$ .

$y = 6 + 0$

Step 1.4.3.2

Add $6$ and $0$ .

$y = 6$

Step 1.4.4

If the table has a linear function rule, $y = q (x)$ for the corresponding $x$ value, $x = 3$ . This check passes since $y = 6$ and $q (x) = 6$ .

$6 = 6$

Step 1.4.5

Calculate the value of $y$ when $a = 2$ , $b = 0$ , and $x = 8$ .

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Step 1.4.5.1

Multiply $2$ by $8$ .

$y = 16 + 0$

Step 1.4.5.2

Add $16$ and $0$ .

$y = 16$

Step 1.4.6

If the table has a linear function rule, $y = q (x)$ for the corresponding $x$ value, $x = 8$ . This check passes since $y = 16$ and $q (x) = 16$ .

$16 = 16$

Step 1.4.7

Calculate the value of $y$ when $a = 2$ , $b = 0$ , and $x = 12$ .

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Step 1.4.7.1

Multiply $2$ by $12$ .

$y = 24 + 0$

Step 1.4.7.2

Add $24$ and $0$ .

$y = 24$

Step 1.4.8

If the table has a linear function rule, $y = q (x)$ for the corresponding $x$ value, $x = 12$ . This check passes since $y = 24$ and $q (x) = 24$ .

$24 = 24$

Step 1.4.9

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 $y = 2 x$ .

$y = 2 x$

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