Algebra Examples

$\frac{11 y + 13}{3 y - 3} + \frac{15 y + 17}{4 - 4 y}$

Step 1

Simplify each term.

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Step 1.1

Factor $3$ out of $3 y - 3$ .

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Step 1.1.1

Factor $3$ out of $3 y$ .

$\frac{11 y + 13}{3 (y) - 3} + \frac{15 y + 17}{4 - 4 y}$

Step 1.1.2

Factor $3$ out of $- 3$ .

$\frac{11 y + 13}{3 (y) + 3 (- 1)} + \frac{15 y + 17}{4 - 4 y}$

Step 1.1.3

Factor $3$ out of $3 (y) + 3 (- 1)$ .

$\frac{11 y + 13}{3 (y - 1)} + \frac{15 y + 17}{4 - 4 y}$

Step 1.2

Factor $4$ out of $4 - 4 y$ .

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Step 1.2.1

Factor $4$ out of $4$ .

$\frac{11 y + 13}{3 (y - 1)} + \frac{15 y + 17}{4 (1) - 4 y}$

Step 1.2.2

Factor $4$ out of $- 4 y$ .

$\frac{11 y + 13}{3 (y - 1)} + \frac{15 y + 17}{4 (1) + 4 (- y)}$

Step 1.2.3

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

$\frac{11 y + 13}{3 (y - 1)} + \frac{15 y + 17}{4 (1 - y)}$

Step 2

Simplify with factoring out.

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Step 2.1

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

$\frac{11 y + 13}{3 (y - 1)} + \frac{15 y + 17}{4 (- 1 (- 1) - y)}$

Step 2.2

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

$\frac{11 y + 13}{3 (y - 1)} + \frac{15 y + 17}{4 (- 1 (- 1) - (y))}$

Step 2.3

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

$\frac{11 y + 13}{3 (y - 1)} + \frac{15 y + 17}{4 (- 1 (- 1 + y))}$

Step 2.4

Simplify the expression.

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Step 2.4.1

Move a negative from the denominator of $\frac{15 y + 17}{4 (- 1 (- 1 + y))}$ to the numerator.

$\frac{11 y + 13}{3 (y - 1)} + \frac{- (15 y + 17)}{4 (- 1 + y)}$

Step 2.4.2

Reorder terms.

$\frac{11 y + 13}{3 (y - 1)} + \frac{- (15 y + 17)}{4 (y - 1)}$

Step 3

To write $\frac{11 y + 13}{3 (y - 1)}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{11 y + 13}{3 (y - 1)} \cdot \frac{4}{4} + \frac{- (15 y + 17)}{4 (y - 1)}$

Step 4

To write $\frac{- (15 y + 17)}{4 (y - 1)}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{11 y + 13}{3 (y - 1)} \cdot \frac{4}{4} + \frac{- (15 y + 17)}{4 (y - 1)} \cdot \frac{3}{3}$

Step 5

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

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Step 5.1

Multiply $\frac{11 y + 13}{3 (y - 1)}$ by $\frac{4}{4}$ .

$\frac{(11 y + 13) \cdot 4}{3 (y - 1) \cdot 4} + \frac{- (15 y + 17)}{4 (y - 1)} \cdot \frac{3}{3}$

Step 5.2

Multiply $4$ by $3$ .

$\frac{(11 y + 13) \cdot 4}{12 (y - 1)} + \frac{- (15 y + 17)}{4 (y - 1)} \cdot \frac{3}{3}$

Step 5.3

Multiply $\frac{- (15 y + 17)}{4 (y - 1)}$ by $\frac{3}{3}$ .

$\frac{(11 y + 13) \cdot 4}{12 (y - 1)} + \frac{- (15 y + 17) \cdot 3}{4 (y - 1) \cdot 3}$

Step 5.4

Multiply $3$ by $4$ .

$\frac{(11 y + 13) \cdot 4}{12 (y - 1)} + \frac{- (15 y + 17) \cdot 3}{12 (y - 1)}$

Step 6

Combine the numerators over the common denominator.

$\frac{(11 y + 13) \cdot 4 - (15 y + 17) \cdot 3}{12 (y - 1)}$

Step 7

Simplify the numerator.

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Step 7.1

Apply the distributive property.

$\frac{11 y \cdot 4 + 13 \cdot 4 - (15 y + 17) \cdot 3}{12 (y - 1)}$

Step 7.2

Multiply $4$ by $11$ .

$\frac{44 y + 13 \cdot 4 - (15 y + 17) \cdot 3}{12 (y - 1)}$

Step 7.3

Multiply $13$ by $4$ .

$\frac{44 y + 52 - (15 y + 17) \cdot 3}{12 (y - 1)}$

Step 7.4

Apply the distributive property.

$\frac{44 y + 52 + (- (15 y) - 1 \cdot 17) \cdot 3}{12 (y - 1)}$

Step 7.5

Multiply $15$ by $- 1$ .

$\frac{44 y + 52 + (- 15 y - 1 \cdot 17) \cdot 3}{12 (y - 1)}$

Step 7.6

Multiply $- 1$ by $17$ .

$\frac{44 y + 52 + (- 15 y - 17) \cdot 3}{12 (y - 1)}$

Step 7.7

Apply the distributive property.

$\frac{44 y + 52 - 15 y \cdot 3 - 17 \cdot 3}{12 (y - 1)}$

Step 7.8

Multiply $3$ by $- 15$ .

$\frac{44 y + 52 - 45 y - 17 \cdot 3}{12 (y - 1)}$

Step 7.9

Multiply $- 17$ by $3$ .

$\frac{44 y + 52 - 45 y - 51}{12 (y - 1)}$

Step 7.10

Subtract $45 y$ from $44 y$ .

$\frac{- y + 52 - 51}{12 (y - 1)}$

Step 7.11

Subtract $51$ from $52$ .

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

Step 8

Reduce the expression by cancelling the common factors.

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Step 8.1

Cancel the common factor of $- y + 1$ and $y - 1$ .

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Step 8.1.1

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

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

Step 8.1.2

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

$\frac{- (y) - 1 (- 1)}{12 (y - 1)}$

Step 8.1.3

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

$\frac{- (y - 1)}{12 (y - 1)}$

Step 8.1.4

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

$\frac{- 1 (y - 1)}{12 (y - 1)}$

Step 8.1.5

Cancel the common factor.

$\frac{- 1 (y - 1)}{12 (y - 1)}$

Step 8.1.6

Rewrite the expression.

$\frac{- 1}{12}$

Step 8.2

Move the negative in front of the fraction.

$- \frac{1}{12}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{12}$

Decimal Form:

$- 0.08\overline{3}$