Pre-Algebra Examples

$\frac{1}{2} \cdot (y + 8) + 5 = \frac{1}{4} \cdot (8 y - 14) - 7$

Step 1

Simplify $\frac{1}{2} \cdot (y + 8) + 5$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$\frac{1}{2} y + \frac{1}{2} \cdot 8 + 5 = \frac{1}{4} \cdot (8 y - 14) - 7$

Step 1.1.2

Combine $\frac{1}{2}$ and $y$ .

$\frac{y}{2} + \frac{1}{2} \cdot 8 + 5 = \frac{1}{4} \cdot (8 y - 14) - 7$

Step 1.1.3

Cancel the common factor of $2$ .

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Step 1.1.3.1

Factor $2$ out of $8$ .

$\frac{y}{2} + \frac{1}{2} \cdot (2 (4)) + 5 = \frac{1}{4} \cdot (8 y - 14) - 7$

Step 1.1.3.2

Cancel the common factor.

$\frac{y}{2} + \frac{1}{2} \cdot (2 \cdot 4) + 5 = \frac{1}{4} \cdot (8 y - 14) - 7$

Step 1.1.3.3

Rewrite the expression.

$\frac{y}{2} + 4 + 5 = \frac{1}{4} \cdot (8 y - 14) - 7$

Step 1.2

Add $4$ and $5$ .

$\frac{y}{2} + 9 = \frac{1}{4} \cdot (8 y - 14) - 7$

Step 2

Simplify $\frac{1}{4} \cdot (8 y - 14) - 7$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$\frac{y}{2} + 9 = \frac{1}{4} (8 y) + \frac{1}{4} \cdot - 14 - 7$

Step 2.1.2

Cancel the common factor of $4$ .

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Step 2.1.2.1

Factor $4$ out of $8 y$ .

$\frac{y}{2} + 9 = \frac{1}{4} (4 (2 y)) + \frac{1}{4} \cdot - 14 - 7$

Step 2.1.2.2

Cancel the common factor.

$\frac{y}{2} + 9 = \frac{1}{4} (4 (2 y)) + \frac{1}{4} \cdot - 14 - 7$

Step 2.1.2.3

Rewrite the expression.

$\frac{y}{2} + 9 = 2 y + \frac{1}{4} \cdot - 14 - 7$

Step 2.1.3

Cancel the common factor of $2$ .

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Step 2.1.3.1

Factor $2$ out of $4$ .

$\frac{y}{2} + 9 = 2 y + \frac{1}{2 (2)} \cdot - 14 - 7$

Step 2.1.3.2

Factor $2$ out of $- 14$ .

$\frac{y}{2} + 9 = 2 y + \frac{1}{2 \cdot 2} \cdot (2 \cdot - 7) - 7$

Step 2.1.3.3

Cancel the common factor.

$\frac{y}{2} + 9 = 2 y + \frac{1}{2 \cdot 2} \cdot (2 \cdot - 7) - 7$

Step 2.1.3.4

Rewrite the expression.

$\frac{y}{2} + 9 = 2 y + \frac{1}{2} \cdot - 7 - 7$

Step 2.1.4

Combine $\frac{1}{2}$ and $- 7$ .

$\frac{y}{2} + 9 = 2 y + \frac{- 7}{2} - 7$

Step 2.1.5

Move the negative in front of the fraction.

$\frac{y}{2} + 9 = 2 y - \frac{7}{2} - 7$

Step 2.2

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{y}{2} + 9 = 2 y - \frac{7}{2} - 7 \cdot \frac{2}{2}$

Step 2.3

Combine $- 7$ and $\frac{2}{2}$ .

$\frac{y}{2} + 9 = 2 y - \frac{7}{2} + \frac{- 7 \cdot 2}{2}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{y}{2} + 9 = 2 y + \frac{- 7 - 7 \cdot 2}{2}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $- 7$ by $2$ .

$\frac{y}{2} + 9 = 2 y + \frac{- 7 - 14}{2}$

Step 2.5.2

Subtract $14$ from $- 7$ .

$\frac{y}{2} + 9 = 2 y + \frac{- 21}{2}$

Step 2.6

Move the negative in front of the fraction.

$\frac{y}{2} + 9 = 2 y - \frac{21}{2}$

Step 3

Move all terms containing $y$ to the left side of the equation.

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Step 3.1

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

$\frac{y}{2} + 9 - 2 y = - \frac{21}{2}$

Step 3.2

To write $- 2 y$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{y}{2} - 2 y \cdot \frac{2}{2} + 9 = - \frac{21}{2}$

Step 3.3

Combine $- 2 y$ and $\frac{2}{2}$ .

$\frac{y}{2} + \frac{- 2 y \cdot 2}{2} + 9 = - \frac{21}{2}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{y - 2 y \cdot 2}{2} + 9 = - \frac{21}{2}$

Step 3.5

Simplify each term.

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

Factor $y$ out of $y - 2 y \cdot 2$ .

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Step 3.5.1.1.1

Raise $y$ to the power of $1$ .

$\frac{y^{1} - 2 y \cdot 2}{2} + 9 = - \frac{21}{2}$

Step 3.5.1.1.2

Factor $y$ out of $y^{1}$ .

$\frac{y \cdot 1 - 2 y \cdot 2}{2} + 9 = - \frac{21}{2}$

Step 3.5.1.1.3

Factor $y$ out of $- 2 y \cdot 2$ .

$\frac{y \cdot 1 + y (- 2 \cdot 2)}{2} + 9 = - \frac{21}{2}$

Step 3.5.1.1.4

Factor $y$ out of $y \cdot 1 + y (- 2 \cdot 2)$ .

$\frac{y (1 - 2 \cdot 2)}{2} + 9 = - \frac{21}{2}$

Step 3.5.1.2

Multiply $- 2$ by $2$ .

$\frac{y (1 - 4)}{2} + 9 = - \frac{21}{2}$

Step 3.5.1.3

Subtract $4$ from $1$ .

$\frac{y \cdot - 3}{2} + 9 = - \frac{21}{2}$

Step 3.5.2

Move $- 3$ to the left of $y$ .

$\frac{- 3 \cdot y}{2} + 9 = - \frac{21}{2}$

Step 3.5.3

Move the negative in front of the fraction.

$- \frac{3 y}{2} + 9 = - \frac{21}{2}$

Step 4

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

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Step 4.1

Subtract $9$ from both sides of the equation.

$- \frac{3 y}{2} = - \frac{21}{2} - 9$

Step 4.2

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{3 y}{2} = - \frac{21}{2} - 9 \cdot \frac{2}{2}$

Step 4.3

Combine $- 9$ and $\frac{2}{2}$ .

$- \frac{3 y}{2} = - \frac{21}{2} + \frac{- 9 \cdot 2}{2}$

Step 4.4

Combine the numerators over the common denominator.

$- \frac{3 y}{2} = \frac{- 21 - 9 \cdot 2}{2}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $- 9$ by $2$ .

$- \frac{3 y}{2} = \frac{- 21 - 18}{2}$

Step 4.5.2

Subtract $18$ from $- 21$ .

$- \frac{3 y}{2} = \frac{- 39}{2}$

Step 4.6

Move the negative in front of the fraction.

$- \frac{3 y}{2} = - \frac{39}{2}$

Step 5

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$- 3 y = - 39$

Step 6

Divide each term in $- 3 y = - 39$ by $- 3$ and simplify.

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Step 6.1

Divide each term in $- 3 y = - 39$ by $- 3$ .

$\frac{- 3 y}{- 3} = \frac{- 39}{- 3}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $- 3$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{- 3 y}{- 3} = \frac{- 39}{- 3}$

Step 6.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 39}{- 3}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $- 39$ by $- 3$ .

$y = 13$