Pre-Algebra Examples

$\frac{2 y}{3} = \frac{1}{2} \cdot (- \frac{3}{2} + \frac{y}{3})$

Step 1

Multiply both sides by $3$ .

$\frac{2 y}{3} \cdot 3 = \frac{1}{2} \cdot (- \frac{3}{2} + \frac{y}{3}) \cdot 3$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Cancel the common factor of $3$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{2 y}{3} \cdot 3 = \frac{1}{2} \cdot (- \frac{3}{2} + \frac{y}{3}) \cdot 3$

Step 2.1.1.2

Rewrite the expression.

$2 y = \frac{1}{2} \cdot (- \frac{3}{2} + \frac{y}{3}) \cdot 3$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $\frac{1}{2} \cdot (- \frac{3}{2} + \frac{y}{3}) \cdot 3$ .

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Step 2.2.1.1

Apply the distributive property.

$2 y = (\frac{1}{2} (- \frac{3}{2}) + \frac{1}{2} \cdot \frac{y}{3}) \cdot 3$

Step 2.2.1.2

Multiply $\frac{1}{2} (- \frac{3}{2})$ .

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Step 2.2.1.2.1

Multiply $\frac{1}{2}$ by $\frac{3}{2}$ .

$2 y = (- \frac{3}{2 \cdot 2} + \frac{1}{2} \cdot \frac{y}{3}) \cdot 3$

Step 2.2.1.2.2

Multiply $2$ by $2$ .

$2 y = (- \frac{3}{4} + \frac{1}{2} \cdot \frac{y}{3}) \cdot 3$

Step 2.2.1.3

Multiply $\frac{1}{2} \cdot \frac{y}{3}$ .

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Step 2.2.1.3.1

Multiply $\frac{1}{2}$ by $\frac{y}{3}$ .

$2 y = (- \frac{3}{4} + \frac{y}{2 \cdot 3}) \cdot 3$

Step 2.2.1.3.2

Multiply $2$ by $3$ .

$2 y = (- \frac{3}{4} + \frac{y}{6}) \cdot 3$

Step 2.2.1.4

Apply the distributive property.

$2 y = - \frac{3}{4} \cdot 3 + \frac{y}{6} \cdot 3$

Step 2.2.1.5

Multiply $- \frac{3}{4} \cdot 3$ .

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Step 2.2.1.5.1

Multiply $3$ by $- 1$ .

$2 y = - 3 (\frac{3}{4}) + \frac{y}{6} \cdot 3$

Step 2.2.1.5.2

Combine $- 3$ and $\frac{3}{4}$ .

$2 y = \frac{- 3 \cdot 3}{4} + \frac{y}{6} \cdot 3$

Step 2.2.1.5.3

Multiply $- 3$ by $3$ .

$2 y = \frac{- 9}{4} + \frac{y}{6} \cdot 3$

Step 2.2.1.6

Cancel the common factor of $3$ .

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Step 2.2.1.6.1

Factor $3$ out of $6$ .

$2 y = \frac{- 9}{4} + \frac{y}{3 (2)} \cdot 3$

Step 2.2.1.6.2

Cancel the common factor.

$2 y = \frac{- 9}{4} + \frac{y}{3 \cdot 2} \cdot 3$

Step 2.2.1.6.3

Rewrite the expression.

$2 y = \frac{- 9}{4} + \frac{y}{2}$

Step 2.2.1.7

Simplify the expression.

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Step 2.2.1.7.1

Move the negative in front of the fraction.

$2 y = - \frac{9}{4} + \frac{y}{2}$

Step 2.2.1.7.2

Reorder $- \frac{9}{4}$ and $\frac{y}{2}$ .

$2 y = \frac{y}{2} - \frac{9}{4}$

Step 3

Solve for $y$ .

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Step 3.1

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

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Step 3.1.1

Subtract $\frac{y}{2}$ from both sides of the equation.

$2 y - \frac{y}{2} = - \frac{9}{4}$

Step 3.1.2

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

$2 y \cdot \frac{2}{2} - \frac{y}{2} = - \frac{9}{4}$

Step 3.1.3

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

$\frac{2 y \cdot 2}{2} - \frac{y}{2} = - \frac{9}{4}$

Step 3.1.4

Combine the numerators over the common denominator.

$\frac{2 y \cdot 2 - y}{2} = - \frac{9}{4}$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

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

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Step 3.1.5.1.1

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

$\frac{y (2 \cdot 2) - y}{2} = - \frac{9}{4}$

Step 3.1.5.1.2

Factor $y$ out of $- y$ .

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

Step 3.1.5.1.3

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

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

Step 3.1.5.2

Multiply $2$ by $2$ .

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

Step 3.1.5.3

Subtract $1$ from $4$ .

$\frac{y \cdot 3}{2} = - \frac{9}{4}$

Step 3.1.6

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

$\frac{3 y}{2} = - \frac{9}{4}$

Step 3.2

Multiply both sides of the equation by $\frac{2}{3}$ .

$\frac{2}{3} \cdot \frac{3 y}{2} = \frac{2}{3} (- \frac{9}{4})$

Step 3.3

Simplify both sides of the equation.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify $\frac{2}{3} \cdot \frac{3 y}{2}$ .

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Step 3.3.1.1.1

Cancel the common factor of $2$ .

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Step 3.3.1.1.1.1

Cancel the common factor.

$\frac{2}{3} \cdot \frac{3 y}{2} = \frac{2}{3} (- \frac{9}{4})$

Step 3.3.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 y) = \frac{2}{3} (- \frac{9}{4})$

Step 3.3.1.1.2

Cancel the common factor of $3$ .

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Step 3.3.1.1.2.1

Factor $3$ out of $3 y$ .

$\frac{1}{3} (3 (y)) = \frac{2}{3} (- \frac{9}{4})$

Step 3.3.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 y) = \frac{2}{3} (- \frac{9}{4})$

Step 3.3.1.1.2.3

Rewrite the expression.

$y = \frac{2}{3} (- \frac{9}{4})$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $\frac{2}{3} (- \frac{9}{4})$ .

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Step 3.3.2.1.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1.1

Move the leading negative in $- \frac{9}{4}$ into the numerator.

$y = \frac{2}{3} \cdot \frac{- 9}{4}$

Step 3.3.2.1.1.2

Factor $2$ out of $4$ .

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

Step 3.3.2.1.1.3

Cancel the common factor.

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

Step 3.3.2.1.1.4

Rewrite the expression.

$y = \frac{1}{3} \cdot \frac{- 9}{2}$

Step 3.3.2.1.2

Cancel the common factor of $3$ .

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Step 3.3.2.1.2.1

Factor $3$ out of $- 9$ .

$y = \frac{1}{3} \cdot \frac{3 (- 3)}{2}$

Step 3.3.2.1.2.2

Cancel the common factor.

$y = \frac{1}{3} \cdot \frac{3 \cdot - 3}{2}$

Step 3.3.2.1.2.3

Rewrite the expression.

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

Step 3.3.2.1.3

Move the negative in front of the fraction.

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$y = - 1.5$

Mixed Number Form:

$y = - 1 \frac{1}{2}$