Algebra Examples

$\frac{y + 5}{2} - \frac{y - 2}{4} = \frac{y + 7}{3} + 1$

Step 1

Simplify $\frac{y + 5}{2} - \frac{y - 2}{4}$ .

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Step 1.1

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

$\frac{y + 5}{2} \cdot \frac{2}{2} - \frac{y - 2}{4} = \frac{y + 7}{3} + 1$

Step 1.2

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.1

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

$\frac{(y + 5) \cdot 2}{2 \cdot 2} - \frac{y - 2}{4} = \frac{y + 7}{3} + 1$

Step 1.2.2

Multiply $2$ by $2$ .

$\frac{(y + 5) \cdot 2}{4} - \frac{y - 2}{4} = \frac{y + 7}{3} + 1$

Step 1.3

Combine the numerators over the common denominator.

$\frac{(y + 5) \cdot 2 - (y - 2)}{4} = \frac{y + 7}{3} + 1$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Apply the distributive property.

$\frac{y \cdot 2 + 5 \cdot 2 - (y - 2)}{4} = \frac{y + 7}{3} + 1$

Step 1.4.2

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

$\frac{2 \cdot y + 5 \cdot 2 - (y - 2)}{4} = \frac{y + 7}{3} + 1$

Step 1.4.3

Multiply $5$ by $2$ .

$\frac{2 \cdot y + 10 - (y - 2)}{4} = \frac{y + 7}{3} + 1$

Step 1.4.4

Apply the distributive property.

$\frac{2 y + 10 - y - - 2}{4} = \frac{y + 7}{3} + 1$

Step 1.4.5

Multiply $- 1$ by $- 2$ .

$\frac{2 y + 10 - y + 2}{4} = \frac{y + 7}{3} + 1$

Step 1.4.6

Subtract $y$ from $2 y$ .

$\frac{y + 10 + 2}{4} = \frac{y + 7}{3} + 1$

Step 1.4.7

Add $10$ and $2$ .

$\frac{y + 12}{4} = \frac{y + 7}{3} + 1$

Step 2

Simplify $\frac{y + 7}{3} + 1$ .

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Step 2.1

Write $1$ as a fraction with a common denominator.

$\frac{y + 12}{4} = \frac{y + 7}{3} + \frac{3}{3}$

Step 2.2

Combine the numerators over the common denominator.

$\frac{y + 12}{4} = \frac{y + 7 + 3}{3}$

Step 2.3

Add $7$ and $3$ .

$\frac{y + 12}{4} = \frac{y + 10}{3}$

Step 3

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

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Step 3.1

Subtract $\frac{y + 10}{3}$ from both sides of the equation.

$\frac{y + 12}{4} - \frac{y + 10}{3} = 0$

Step 3.2

To write $\frac{y + 12}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{y + 12}{4} \cdot \frac{3}{3} - \frac{y + 10}{3} = 0$

Step 3.3

To write $- \frac{y + 10}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{y + 12}{4} \cdot \frac{3}{3} - \frac{y + 10}{3} \cdot \frac{4}{4} = 0$

Step 3.4

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{y + 12}{4}$ by $\frac{3}{3}$ .

$\frac{(y + 12) \cdot 3}{4 \cdot 3} - \frac{y + 10}{3} \cdot \frac{4}{4} = 0$

Step 3.4.2

Multiply $4$ by $3$ .

$\frac{(y + 12) \cdot 3}{12} - \frac{y + 10}{3} \cdot \frac{4}{4} = 0$

Step 3.4.3

Multiply $\frac{y + 10}{3}$ by $\frac{4}{4}$ .

$\frac{(y + 12) \cdot 3}{12} - \frac{(y + 10) \cdot 4}{3 \cdot 4} = 0$

Step 3.4.4

Multiply $3$ by $4$ .

$\frac{(y + 12) \cdot 3}{12} - \frac{(y + 10) \cdot 4}{12} = 0$

Step 3.5

Combine the numerators over the common denominator.

$\frac{(y + 12) \cdot 3 - (y + 10) \cdot 4}{12} = 0$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Apply the distributive property.

$\frac{y \cdot 3 + 12 \cdot 3 - (y + 10) \cdot 4}{12} = 0$

Step 3.6.2

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

$\frac{3 \cdot y + 12 \cdot 3 - (y + 10) \cdot 4}{12} = 0$

Step 3.6.3

Multiply $12$ by $3$ .

$\frac{3 \cdot y + 36 - (y + 10) \cdot 4}{12} = 0$

Step 3.6.4

Apply the distributive property.

$\frac{3 y + 36 + (- y - 1 \cdot 10) \cdot 4}{12} = 0$

Step 3.6.5

Multiply $- 1$ by $10$ .

$\frac{3 y + 36 + (- y - 10) \cdot 4}{12} = 0$

Step 3.6.6

Apply the distributive property.

$\frac{3 y + 36 - y \cdot 4 - 10 \cdot 4}{12} = 0$

Step 3.6.7

Multiply $4$ by $- 1$ .

$\frac{3 y + 36 - 4 y - 10 \cdot 4}{12} = 0$

Step 3.6.8

Multiply $- 10$ by $4$ .

$\frac{3 y + 36 - 4 y - 40}{12} = 0$

Step 3.6.9

Subtract $4 y$ from $3 y$ .

$\frac{- y + 36 - 40}{12} = 0$

Step 3.6.10

Subtract $40$ from $36$ .

$\frac{- y - 4}{12} = 0$

Step 3.7

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

$\frac{- (y) - 4}{12} = 0$

Step 3.8

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{- (y) - 1 (4)}{12} = 0$

Step 3.9

Factor $- 1$ out of $- (y) - 1 (4)$ .

$\frac{- (y + 4)}{12} = 0$

Step 3.10

Rewrite $- (y + 4)$ as $- 1 (y + 4)$ .

$\frac{- 1 (y + 4)}{12} = 0$

Step 3.11

Move the negative in front of the fraction.

$- \frac{y + 4}{12} = 0$

Step 4

Set the numerator equal to zero.

$y + 4 = 0$

Step 5

Subtract $4$ from both sides of the equation.

$y = - 4$