Pre-Algebra Examples

$\begin{array}{c} x & y \\ 2 & 5 \\ 4 & 10 \\ 6 & 15 \\ 8 & 20 \\ 10 & 25 \end{array}$

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 $y = a x + b$ .

$5 = a (2) + b 10 = a (4) + b 15 = a (6) + b 20 = a (8) + b 25 = a (10) + b$

Step 1.3

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

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Step 1.3.1

Solve for $b$ in $5 = a (2) + b$ .

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Step 1.3.1.1

Rewrite the equation as $a (2) + b = 5$ .

$a (2) + b = 5$

$10 = a (4) + b$

$15 = a (6) + b$

$20 = a (8) + b$

$25 = a (10) + b$

Step 1.3.1.2

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

$2 a + b = 5$

$10 = a (4) + b$

$15 = a (6) + b$

$20 = a (8) + b$

$25 = a (10) + b$

Step 1.3.1.3

Subtract $2 a$ from both sides of the equation.

$b = 5 - 2 a$

$10 = a (4) + b$

$15 = a (6) + b$

$20 = a (8) + b$

$25 = a (10) + b$

$b = 5 - 2 a$

$10 = a (4) + b$

$15 = a (6) + b$

$20 = a (8) + b$

$25 = a (10) + b$

Step 1.3.2

Replace all occurrences of $b$ with $5 - 2 a$ in each equation.

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Step 1.3.2.1

Replace all occurrences of $b$ in $10 = a (4) + b$ with $5 - 2 a$ .

$10 = a (4) + 5 - 2 a$

$b = 5 - 2 a$

$15 = a (6) + b$

$20 = a (8) + b$

$25 = a (10) + b$

Step 1.3.2.2

Simplify $10 = a (4) + 5 - 2 a$ .

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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.

$10 = a (4) + 5 - 2 a$

$b = 5 - 2 a$

$15 = a (6) + b$

$20 = a (8) + b$

$25 = a (10) + b$

$10 = a (4) + 5 - 2 a$

$b = 5 - 2 a$

$15 = a (6) + b$

$20 = a (8) + b$

$25 = a (10) + b$

Step 1.3.2.2.2

Simplify the right side.

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Step 1.3.2.2.2.1

Simplify $a (4) + 5 - 2 a$ .

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Step 1.3.2.2.2.1.1

Move $4$ to the left of $a$ .

$10 = 4 a + 5 - 2 a$

$b = 5 - 2 a$

$15 = a (6) + b$

$20 = a (8) + b$

$25 = a (10) + b$

Step 1.3.2.2.2.1.2

Subtract $2 a$ from $4 a$ .

$10 = 2 a + 5$

$b = 5 - 2 a$

$15 = a (6) + b$

$20 = a (8) + b$

$25 = a (10) + b$

$10 = 2 a + 5$

$b = 5 - 2 a$

$15 = a (6) + b$

$20 = a (8) + b$

$25 = a (10) + b$

$10 = 2 a + 5$

$b = 5 - 2 a$

$15 = a (6) + b$

$20 = a (8) + b$

$25 = a (10) + b$

$10 = 2 a + 5$

$b = 5 - 2 a$

$15 = a (6) + b$

$20 = a (8) + b$

$25 = a (10) + b$

Step 1.3.2.3

Replace all occurrences of $b$ in $15 = a (6) + b$ with $5 - 2 a$ .

$15 = a (6) + 5 - 2 a$

$10 = 2 a + 5$

$b = 5 - 2 a$

$20 = a (8) + b$

$25 = a (10) + b$

Step 1.3.2.4

Simplify $15 = a (6) + 5 - 2 a$ .

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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.

$15 = a (6) + 5 - 2 a$

$10 = 2 a + 5$

$b = 5 - 2 a$

$20 = a (8) + b$

$25 = a (10) + b$

$15 = a (6) + 5 - 2 a$

$10 = 2 a + 5$

$b = 5 - 2 a$

$20 = a (8) + b$

$25 = a (10) + b$

Step 1.3.2.4.2

Simplify the right side.

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Step 1.3.2.4.2.1

Simplify $a (6) + 5 - 2 a$ .

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Step 1.3.2.4.2.1.1

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

$15 = 6 a + 5 - 2 a$

$10 = 2 a + 5$

$b = 5 - 2 a$

$20 = a (8) + b$

$25 = a (10) + b$

Step 1.3.2.4.2.1.2

Subtract $2 a$ from $6 a$ .

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

$20 = a (8) + b$

$25 = a (10) + b$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

$20 = a (8) + b$

$25 = a (10) + b$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

$20 = a (8) + b$

$25 = a (10) + b$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

$20 = a (8) + b$

$25 = a (10) + b$

Step 1.3.2.5

Replace all occurrences of $b$ in $20 = a (8) + b$ with $5 - 2 a$ .

$20 = a (8) + 5 - 2 a$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

$25 = a (10) + b$

Step 1.3.2.6

Simplify $20 = a (8) + 5 - 2 a$ .

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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.

$20 = a (8) + 5 - 2 a$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

$25 = a (10) + b$

$20 = a (8) + 5 - 2 a$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

$25 = a (10) + b$

Step 1.3.2.6.2

Simplify the right side.

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Step 1.3.2.6.2.1

Simplify $a (8) + 5 - 2 a$ .

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Step 1.3.2.6.2.1.1

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

$20 = 8 a + 5 - 2 a$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

$25 = a (10) + b$

Step 1.3.2.6.2.1.2

Subtract $2 a$ from $8 a$ .

$20 = 6 a + 5$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

$25 = a (10) + b$

$20 = 6 a + 5$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

$25 = a (10) + b$

$20 = 6 a + 5$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

$25 = a (10) + b$

$20 = 6 a + 5$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

$25 = a (10) + b$

Step 1.3.2.7

Replace all occurrences of $b$ in $25 = a (10) + b$ with $5 - 2 a$ .

$25 = a (10) + 5 - 2 a$

$20 = 6 a + 5$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

Step 1.3.2.8

Simplify $25 = a (10) + 5 - 2 a$ .

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Step 1.3.2.8.1

Simplify the left side.

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Step 1.3.2.8.1.1

Remove parentheses.

$25 = a (10) + 5 - 2 a$

$20 = 6 a + 5$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

$25 = a (10) + 5 - 2 a$

$20 = 6 a + 5$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

Step 1.3.2.8.2

Simplify the right side.

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Step 1.3.2.8.2.1

Simplify $a (10) + 5 - 2 a$ .

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Step 1.3.2.8.2.1.1

Move $10$ to the left of $a$ .

$25 = 10 a + 5 - 2 a$

$20 = 6 a + 5$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

Step 1.3.2.8.2.1.2

Subtract $2 a$ from $10 a$ .

$25 = 8 a + 5$

$20 = 6 a + 5$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

$25 = 8 a + 5$

$20 = 6 a + 5$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

$25 = 8 a + 5$

$20 = 6 a + 5$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

$25 = 8 a + 5$

$20 = 6 a + 5$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

$25 = 8 a + 5$

$20 = 6 a + 5$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

Step 1.3.3

Solve for $a$ in $25 = 8 a + 5$ .

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Step 1.3.3.1

Rewrite the equation as $8 a + 5 = 25$ .

$8 a + 5 = 25$

$20 = 6 a + 5$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

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 $5$ from both sides of the equation.

$8 a = 25 - 5$

$20 = 6 a + 5$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

Step 1.3.3.2.2

Subtract $5$ from $25$ .

$8 a = 20$

$20 = 6 a + 5$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

$8 a = 20$

$20 = 6 a + 5$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

Step 1.3.3.3

Divide each term in $8 a = 20$ by $8$ and simplify.

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Step 1.3.3.3.1

Divide each term in $8 a = 20$ by $8$ .

$\frac{8 a}{8} = \frac{20}{8}$

$20 = 6 a + 5$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

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 $8$ .

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Step 1.3.3.3.2.1.1

Cancel the common factor.

$\frac{8 a}{8} = \frac{20}{8}$

$20 = 6 a + 5$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

Step 1.3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{20}{8}$

$20 = 6 a + 5$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

$a = \frac{20}{8}$

$20 = 6 a + 5$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

$a = \frac{20}{8}$

$20 = 6 a + 5$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

Step 1.3.3.3.3

Simplify the right side.

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Step 1.3.3.3.3.1

Cancel the common factor of $20$ and $8$ .

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Step 1.3.3.3.3.1.1

Factor $4$ out of $20$ .

$a = \frac{4 (5)}{8}$

$20 = 6 a + 5$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

Step 1.3.3.3.3.1.2

Cancel the common factors.

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Step 1.3.3.3.3.1.2.1

Factor $4$ out of $8$ .

$a = \frac{4 \cdot 5}{4 \cdot 2}$

$20 = 6 a + 5$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

Step 1.3.3.3.3.1.2.2

Cancel the common factor.

$a = \frac{4 \cdot 5}{4 \cdot 2}$

$20 = 6 a + 5$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

Step 1.3.3.3.3.1.2.3

Rewrite the expression.

$a = \frac{5}{2}$

$20 = 6 a + 5$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

$a = \frac{5}{2}$

$20 = 6 a + 5$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

$a = \frac{5}{2}$

$20 = 6 a + 5$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

$a = \frac{5}{2}$

$20 = 6 a + 5$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

$a = \frac{5}{2}$

$20 = 6 a + 5$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

$a = \frac{5}{2}$

$20 = 6 a + 5$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

Step 1.3.4

Replace all occurrences of $a$ with $\frac{5}{2}$ in each equation.

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Step 1.3.4.1

Replace all occurrences of $a$ in $20 = 6 a + 5$ with $\frac{5}{2}$ .

$20 = 6 (\frac{5}{2}) + 5$

$a = \frac{5}{2}$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

Step 1.3.4.2

Simplify the right side.

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Step 1.3.4.2.1

Simplify $6 (\frac{5}{2}) + 5$ .

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Step 1.3.4.2.1.1

Simplify each term.

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Step 1.3.4.2.1.1.1

Cancel the common factor of $2$ .

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Step 1.3.4.2.1.1.1.1

Factor $2$ out of $6$ .

$20 = 2 (3) (\frac{5}{2}) + 5$

$a = \frac{5}{2}$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

Step 1.3.4.2.1.1.1.2

Cancel the common factor.

$20 = 2 \cdot (3 (\frac{5}{2})) + 5$

$a = \frac{5}{2}$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

Step 1.3.4.2.1.1.1.3

Rewrite the expression.

$20 = 3 \cdot 5 + 5$

$a = \frac{5}{2}$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

$20 = 3 \cdot 5 + 5$

$a = \frac{5}{2}$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

Step 1.3.4.2.1.1.2

Multiply $3$ by $5$ .

$20 = 15 + 5$

$a = \frac{5}{2}$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

$20 = 15 + 5$

$a = \frac{5}{2}$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

Step 1.3.4.2.1.2

Add $15$ and $5$ .

$20 = 20$

$a = \frac{5}{2}$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

$20 = 20$

$a = \frac{5}{2}$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

$20 = 20$

$a = \frac{5}{2}$

$15 = 4 a + 5$

$10 = 2 a + 5$

$b = 5 - 2 a$

Step 1.3.4.3

Replace all occurrences of $a$ in $15 = 4 a + 5$ with $\frac{5}{2}$ .

$15 = 4 (\frac{5}{2}) + 5$

$20 = 20$

$a = \frac{5}{2}$

$10 = 2 a + 5$

$b = 5 - 2 a$

Step 1.3.4.4

Simplify the right side.

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Step 1.3.4.4.1

Simplify $4 (\frac{5}{2}) + 5$ .

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Step 1.3.4.4.1.1

Simplify each term.

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Step 1.3.4.4.1.1.1

Cancel the common factor of $2$ .

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Step 1.3.4.4.1.1.1.1

Factor $2$ out of $4$ .

$15 = 2 (2) (\frac{5}{2}) + 5$

$20 = 20$

$a = \frac{5}{2}$

$10 = 2 a + 5$

$b = 5 - 2 a$

Step 1.3.4.4.1.1.1.2

Cancel the common factor.

$15 = 2 \cdot (2 (\frac{5}{2})) + 5$

$20 = 20$

$a = \frac{5}{2}$

$10 = 2 a + 5$

$b = 5 - 2 a$

Step 1.3.4.4.1.1.1.3

Rewrite the expression.

$15 = 2 \cdot 5 + 5$

$20 = 20$

$a = \frac{5}{2}$

$10 = 2 a + 5$

$b = 5 - 2 a$

$15 = 2 \cdot 5 + 5$

$20 = 20$

$a = \frac{5}{2}$

$10 = 2 a + 5$

$b = 5 - 2 a$

Step 1.3.4.4.1.1.2

Multiply $2$ by $5$ .

$15 = 10 + 5$

$20 = 20$

$a = \frac{5}{2}$

$10 = 2 a + 5$

$b = 5 - 2 a$

$15 = 10 + 5$

$20 = 20$

$a = \frac{5}{2}$

$10 = 2 a + 5$

$b = 5 - 2 a$

Step 1.3.4.4.1.2

Add $10$ and $5$ .

$15 = 15$

$20 = 20$

$a = \frac{5}{2}$

$10 = 2 a + 5$

$b = 5 - 2 a$

$15 = 15$

$20 = 20$

$a = \frac{5}{2}$

$10 = 2 a + 5$

$b = 5 - 2 a$

$15 = 15$

$20 = 20$

$a = \frac{5}{2}$

$10 = 2 a + 5$

$b = 5 - 2 a$

Step 1.3.4.5

Replace all occurrences of $a$ in $10 = 2 a + 5$ with $\frac{5}{2}$ .

$10 = 2 (\frac{5}{2}) + 5$

$15 = 15$

$20 = 20$

$a = \frac{5}{2}$

$b = 5 - 2 a$

Step 1.3.4.6

Simplify the right side.

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Step 1.3.4.6.1

Simplify $2 (\frac{5}{2}) + 5$ .

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Step 1.3.4.6.1.1

Cancel the common factor of $2$ .

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Step 1.3.4.6.1.1.1

Cancel the common factor.

$10 = 2 (\frac{5}{2}) + 5$

$15 = 15$

$20 = 20$

$a = \frac{5}{2}$

$b = 5 - 2 a$

Step 1.3.4.6.1.1.2

Rewrite the expression.

$10 = 5 + 5$

$15 = 15$

$20 = 20$

$a = \frac{5}{2}$

$b = 5 - 2 a$

$10 = 5 + 5$

$15 = 15$

$20 = 20$

$a = \frac{5}{2}$

$b = 5 - 2 a$

Step 1.3.4.6.1.2

Add $5$ and $5$ .

$10 = 10$

$15 = 15$

$20 = 20$

$a = \frac{5}{2}$

$b = 5 - 2 a$

$10 = 10$

$15 = 15$

$20 = 20$

$a = \frac{5}{2}$

$b = 5 - 2 a$

$10 = 10$

$15 = 15$

$20 = 20$

$a = \frac{5}{2}$

$b = 5 - 2 a$

Step 1.3.4.7

Replace all occurrences of $a$ in $b = 5 - 2 a$ with $\frac{5}{2}$ .

$b = 5 - 2 (\frac{5}{2})$

$10 = 10$

$15 = 15$

$20 = 20$

$a = \frac{5}{2}$

Step 1.3.4.8

Simplify the right side.

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Step 1.3.4.8.1

Simplify $5 - 2 (\frac{5}{2})$ .

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Step 1.3.4.8.1.1

Simplify each term.

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Step 1.3.4.8.1.1.1

Cancel the common factor of $2$ .

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Step 1.3.4.8.1.1.1.1

Factor $2$ out of $- 2$ .

$b = 5 + 2 (- 1) (\frac{5}{2})$

$10 = 10$

$15 = 15$

$20 = 20$

$a = \frac{5}{2}$

Step 1.3.4.8.1.1.1.2

Cancel the common factor.

$b = 5 + 2 \cdot (- 1 (\frac{5}{2}))$

$10 = 10$

$15 = 15$

$20 = 20$

$a = \frac{5}{2}$

Step 1.3.4.8.1.1.1.3

Rewrite the expression.

$b = 5 - 1 \cdot 5$

$10 = 10$

$15 = 15$

$20 = 20$

$a = \frac{5}{2}$

$b = 5 - 1 \cdot 5$

$10 = 10$

$15 = 15$

$20 = 20$

$a = \frac{5}{2}$

Step 1.3.4.8.1.1.2

Multiply $- 1$ by $5$ .

$b = 5 - 5$

$10 = 10$

$15 = 15$

$20 = 20$

$a = \frac{5}{2}$

$b = 5 - 5$

$10 = 10$

$15 = 15$

$20 = 20$

$a = \frac{5}{2}$

Step 1.3.4.8.1.2

Subtract $5$ from $5$ .

$b = 0$

$10 = 10$

$15 = 15$

$20 = 20$

$a = \frac{5}{2}$

$b = 0$

$10 = 10$

$15 = 15$

$20 = 20$

$a = \frac{5}{2}$

$b = 0$

$10 = 10$

$15 = 15$

$20 = 20$

$a = \frac{5}{2}$

$b = 0$

$10 = 10$

$15 = 15$

$20 = 20$

$a = \frac{5}{2}$

Step 1.3.5

Remove any equations from the system that are always true.

$b = 0$

$a = \frac{5}{2}$

Step 1.3.6

List all of the solutions.

$b = 0, a = \frac{5}{2}$

Step 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 1.4.1

Calculate the value of $y$ when $a = \frac{5}{2}$ , $b = 0$ , and $x = 2$ .

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Step 1.4.1.1

Cancel the common factor of $2$ .

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Step 1.4.1.1.1

Cancel the common factor.

$y = \frac{5}{2} \cdot 2 + 0$

Step 1.4.1.1.2

Rewrite the expression.

$y = 5 + 0$

Step 1.4.1.2

Add $5$ and $0$ .

$y = 5$

Step 1.4.2

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 2$ . This check passes since $y = 5$ and $y = 5$ .

$5 = 5$

Step 1.4.3

Calculate the value of $y$ when $a = \frac{5}{2}$ , $b = 0$ , and $x = 4$ .

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Step 1.4.3.1

Simplify each term.

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Step 1.4.3.1.1

Cancel the common factor of $2$ .

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Step 1.4.3.1.1.1

Factor $2$ out of $4$ .

$y = \frac{5}{2} \cdot (2 (2)) + 0$

Step 1.4.3.1.1.2

Cancel the common factor.

$y = \frac{5}{2} \cdot (2 \cdot 2) + 0$

Step 1.4.3.1.1.3

Rewrite the expression.

$y = 5 \cdot 2 + 0$

Step 1.4.3.1.2

Multiply $5$ by $2$ .

$y = 10 + 0$

Step 1.4.3.2

Add $10$ and $0$ .

$y = 10$

Step 1.4.4

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 4$ . This check passes since $y = 10$ and $y = 10$ .

$10 = 10$

Step 1.4.5

Calculate the value of $y$ when $a = \frac{5}{2}$ , $b = 0$ , and $x = 6$ .

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Step 1.4.5.1

Simplify each term.

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Step 1.4.5.1.1

Cancel the common factor of $2$ .

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Step 1.4.5.1.1.1

Factor $2$ out of $6$ .

$y = \frac{5}{2} \cdot (2 (3)) + 0$

Step 1.4.5.1.1.2

Cancel the common factor.

$y = \frac{5}{2} \cdot (2 \cdot 3) + 0$

Step 1.4.5.1.1.3

Rewrite the expression.

$y = 5 \cdot 3 + 0$

Step 1.4.5.1.2

Multiply $5$ by $3$ .

$y = 15 + 0$

Step 1.4.5.2

Add $15$ and $0$ .

$y = 15$

Step 1.4.6

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 6$ . This check passes since $y = 15$ and $y = 15$ .

$15 = 15$

Step 1.4.7

Calculate the value of $y$ when $a = \frac{5}{2}$ , $b = 0$ , and $x = 8$ .

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Step 1.4.7.1

Simplify each term.

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Step 1.4.7.1.1

Cancel the common factor of $2$ .

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Step 1.4.7.1.1.1

Factor $2$ out of $8$ .

$y = \frac{5}{2} \cdot (2 (4)) + 0$

Step 1.4.7.1.1.2

Cancel the common factor.

$y = \frac{5}{2} \cdot (2 \cdot 4) + 0$

Step 1.4.7.1.1.3

Rewrite the expression.

$y = 5 \cdot 4 + 0$

Step 1.4.7.1.2

Multiply $5$ by $4$ .

$y = 20 + 0$

Step 1.4.7.2

Add $20$ and $0$ .

$y = 20$

Step 1.4.8

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 8$ . This check passes since $y = 20$ and $y = 20$ .

$20 = 20$

Step 1.4.9

Calculate the value of $y$ when $a = \frac{5}{2}$ , $b = 0$ , and $x = 10$ .

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Step 1.4.9.1

Simplify each term.

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Step 1.4.9.1.1

Cancel the common factor of $2$ .

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Step 1.4.9.1.1.1

Factor $2$ out of $10$ .

$y = \frac{5}{2} \cdot (2 (5)) + 0$

Step 1.4.9.1.1.2

Cancel the common factor.

$y = \frac{5}{2} \cdot (2 \cdot 5) + 0$

Step 1.4.9.1.1.3

Rewrite the expression.

$y = 5 \cdot 5 + 0$

Step 1.4.9.1.2

Multiply $5$ by $5$ .

$y = 25 + 0$

Step 1.4.9.2

Add $25$ and $0$ .

$y = 25$

Step 1.4.10

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 10$ . This check passes since $y = 25$ and $y = 25$ .

$25 = 25$

Step 1.4.11

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 $y = \frac{5 x}{2}$ .

$y = \frac{5 x}{2}$