Algebra Examples

$\frac{9 y}{4 y + 5} + \frac{8 y^{2}}{6 y + 5}$

Step 1

To write $\frac{9 y}{4 y + 5}$ as a fraction with a common denominator, multiply by $\frac{6 y + 5}{6 y + 5}$ .

$\frac{9 y}{4 y + 5} \cdot \frac{6 y + 5}{6 y + 5} + \frac{8 y^{2}}{6 y + 5}$

Step 2

To write $\frac{8 y^{2}}{6 y + 5}$ as a fraction with a common denominator, multiply by $\frac{4 y + 5}{4 y + 5}$ .

$\frac{9 y}{4 y + 5} \cdot \frac{6 y + 5}{6 y + 5} + \frac{8 y^{2}}{6 y + 5} \cdot \frac{4 y + 5}{4 y + 5}$

Step 3

Write each expression with a common denominator of $(4 y + 5) (6 y + 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{9 y}{4 y + 5}$ by $\frac{6 y + 5}{6 y + 5}$ .

$\frac{9 y (6 y + 5)}{(4 y + 5) (6 y + 5)} + \frac{8 y^{2}}{6 y + 5} \cdot \frac{4 y + 5}{4 y + 5}$

Step 3.2

Multiply $\frac{8 y^{2}}{6 y + 5}$ by $\frac{4 y + 5}{4 y + 5}$ .

$\frac{9 y (6 y + 5)}{(4 y + 5) (6 y + 5)} + \frac{8 y^{2} (4 y + 5)}{(6 y + 5) (4 y + 5)}$

Step 3.3

Reorder the factors of $(6 y + 5) (4 y + 5)$ .

$\frac{9 y (6 y + 5)}{(4 y + 5) (6 y + 5)} + \frac{8 y^{2} (4 y + 5)}{(4 y + 5) (6 y + 5)}$

Step 4

Combine the numerators over the common denominator.

$\frac{9 y (6 y + 5) + 8 y^{2} (4 y + 5)}{(4 y + 5) (6 y + 5)}$

Step 5

Simplify the numerator.

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Step 5.1

Factor $y$ out of $9 y (6 y + 5) + 8 y^{2} (4 y + 5)$ .

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Step 5.1.1

Factor $y$ out of $9 y (6 y + 5)$ .

$\frac{y (9 (6 y + 5)) + 8 y^{2} (4 y + 5)}{(4 y + 5) (6 y + 5)}$

Step 5.1.2

Factor $y$ out of $8 y^{2} (4 y + 5)$ .

$\frac{y (9 (6 y + 5)) + y (8 y (4 y + 5))}{(4 y + 5) (6 y + 5)}$

Step 5.1.3

Factor $y$ out of $y (9 (6 y + 5)) + y (8 y (4 y + 5))$ .

$\frac{y (9 (6 y + 5) + 8 y (4 y + 5))}{(4 y + 5) (6 y + 5)}$

Step 5.2

Apply the distributive property.

$\frac{y (9 (6 y) + 9 \cdot 5 + 8 y (4 y + 5))}{(4 y + 5) (6 y + 5)}$

Step 5.3

Multiply $6$ by $9$ .

$\frac{y (54 y + 9 \cdot 5 + 8 y (4 y + 5))}{(4 y + 5) (6 y + 5)}$

Step 5.4

Multiply $9$ by $5$ .

$\frac{y (54 y + 45 + 8 y (4 y + 5))}{(4 y + 5) (6 y + 5)}$

Step 5.5

Apply the distributive property.

$\frac{y (54 y + 45 + 8 y (4 y) + 8 y \cdot 5)}{(4 y + 5) (6 y + 5)}$

Step 5.6

Rewrite using the commutative property of multiplication.

$\frac{y (54 y + 45 + 8 \cdot 4 y \cdot y + 8 y \cdot 5)}{(4 y + 5) (6 y + 5)}$

Step 5.7

Multiply $5$ by $8$ .

$\frac{y (54 y + 45 + 8 \cdot 4 y \cdot y + 40 y)}{(4 y + 5) (6 y + 5)}$

Step 5.8

Simplify each term.

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Step 5.8.1

Multiply $y$ by $y$ by adding the exponents.

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Step 5.8.1.1

Move $y$ .

$\frac{y (54 y + 45 + 8 \cdot 4 (y \cdot y) + 40 y)}{(4 y + 5) (6 y + 5)}$

Step 5.8.1.2

Multiply $y$ by $y$ .

$\frac{y (54 y + 45 + 8 \cdot 4 y^{2} + 40 y)}{(4 y + 5) (6 y + 5)}$

Step 5.8.2

Multiply $8$ by $4$ .

$\frac{y (54 y + 45 + 32 y^{2} + 40 y)}{(4 y + 5) (6 y + 5)}$

Step 5.9

Add $54 y$ and $40 y$ .

$\frac{y (94 y + 45 + 32 y^{2})}{(4 y + 5) (6 y + 5)}$

Step 5.10

Reorder terms.

$\frac{y (32 y^{2} + 94 y + 45)}{(4 y + 5) (6 y + 5)}$