Finite Math Examples

$\frac{3}{2} y - 4 = \frac{5}{3} \cdot (y + 4) - \frac{2}{3}$

Step 1

Combine $\frac{3}{2}$ and $y$ .

$\frac{3 y}{2} - 4 = \frac{5}{3} \cdot (y + 4) - \frac{2}{3}$

Step 2

Simplify $\frac{5}{3} \cdot (y + 4) - \frac{2}{3}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$\frac{3 y}{2} - 4 = \frac{5}{3} y + \frac{5}{3} \cdot 4 - \frac{2}{3}$

Step 2.1.2

Combine $\frac{5}{3}$ and $y$ .

$\frac{3 y}{2} - 4 = \frac{5 y}{3} + \frac{5}{3} \cdot 4 - \frac{2}{3}$

Step 2.1.3

Multiply $\frac{5}{3} \cdot 4$ .

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Step 2.1.3.1

Combine $\frac{5}{3}$ and $4$ .

$\frac{3 y}{2} - 4 = \frac{5 y}{3} + \frac{5 \cdot 4}{3} - \frac{2}{3}$

Step 2.1.3.2

Multiply $5$ by $4$ .

$\frac{3 y}{2} - 4 = \frac{5 y}{3} + \frac{20}{3} - \frac{2}{3}$

Step 2.2

Combine fractions.

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Step 2.2.1

Combine the numerators over the common denominator.

$\frac{3 y}{2} - 4 = \frac{5 y + 20 - 2}{3}$

Step 2.2.2

Subtract $2$ from $20$ .

$\frac{3 y}{2} - 4 = \frac{5 y + 18}{3}$

Step 3

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

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Step 3.1

Subtract $\frac{5 y + 18}{3}$ from both sides of the equation.

$\frac{3 y}{2} - 4 - \frac{5 y + 18}{3} = 0$

Step 3.2

To write $\frac{3 y}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{3 y}{2} \cdot \frac{3}{3} - \frac{5 y + 18}{3} - 4 = 0$

Step 3.3

To write $- \frac{5 y + 18}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 y}{2} \cdot \frac{3}{3} - \frac{5 y + 18}{3} \cdot \frac{2}{2} - 4 = 0$

Step 3.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

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

$\frac{3 y \cdot 3}{2 \cdot 3} - \frac{5 y + 18}{3} \cdot \frac{2}{2} - 4 = 0$

Step 3.4.2

Multiply $2$ by $3$ .

$\frac{3 y \cdot 3}{6} - \frac{5 y + 18}{3} \cdot \frac{2}{2} - 4 = 0$

Step 3.4.3

Multiply $\frac{5 y + 18}{3}$ by $\frac{2}{2}$ .

$\frac{3 y \cdot 3}{6} - \frac{(5 y + 18) \cdot 2}{3 \cdot 2} - 4 = 0$

Step 3.4.4

Multiply $3$ by $2$ .

$\frac{3 y \cdot 3}{6} - \frac{(5 y + 18) \cdot 2}{6} - 4 = 0$

Step 3.5

Combine the numerators over the common denominator.

$\frac{3 y \cdot 3 - (5 y + 18) \cdot 2}{6} - 4 = 0$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $3$ by $3$ .

$\frac{9 y - (5 y + 18) \cdot 2}{6} - 4 = 0$

Step 3.6.2

Apply the distributive property.

$\frac{9 y + (- (5 y) - 1 \cdot 18) \cdot 2}{6} - 4 = 0$

Step 3.6.3

Multiply $5$ by $- 1$ .

$\frac{9 y + (- 5 y - 1 \cdot 18) \cdot 2}{6} - 4 = 0$

Step 3.6.4

Multiply $- 1$ by $18$ .

$\frac{9 y + (- 5 y - 18) \cdot 2}{6} - 4 = 0$

Step 3.6.5

Apply the distributive property.

$\frac{9 y - 5 y \cdot 2 - 18 \cdot 2}{6} - 4 = 0$

Step 3.6.6

Multiply $2$ by $- 5$ .

$\frac{9 y - 10 y - 18 \cdot 2}{6} - 4 = 0$

Step 3.6.7

Multiply $- 18$ by $2$ .

$\frac{9 y - 10 y - 36}{6} - 4 = 0$

Step 3.6.8

Subtract $10 y$ from $9 y$ .

$\frac{- y - 36}{6} - 4 = 0$

Step 3.7

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{- y - 36}{6} - 4 \cdot \frac{6}{6} = 0$

Step 3.8

Combine $- 4$ and $\frac{6}{6}$ .

$\frac{- y - 36}{6} + \frac{- 4 \cdot 6}{6} = 0$

Step 3.9

Combine the numerators over the common denominator.

$\frac{- y - 36 - 4 \cdot 6}{6} = 0$

Step 3.10

Simplify the numerator.

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Step 3.10.1

Multiply $- 4$ by $6$ .

$\frac{- y - 36 - 24}{6} = 0$

Step 3.10.2

Subtract $24$ from $- 36$ .

$\frac{- y - 60}{6} = 0$

Step 3.11

Factor $- 1$ out of $- y$ .

$\frac{- (y) - 60}{6} = 0$

Step 3.12

Rewrite $- 60$ as $- 1 (60)$ .

$\frac{- (y) - 1 (60)}{6} = 0$

Step 3.13

Factor $- 1$ out of $- (y) - 1 (60)$ .

$\frac{- (y + 60)}{6} = 0$

Step 3.14

Rewrite $- (y + 60)$ as $- 1 (y + 60)$ .

$\frac{- 1 (y + 60)}{6} = 0$

Step 3.15

Move the negative in front of the fraction.

$- \frac{y + 60}{6} = 0$

Step 4

Set the numerator equal to zero.

$y + 60 = 0$

Step 5

Subtract $60$ from both sides of the equation.

$y = - 60$