Algebra Examples

$\frac{x^{2} - 9}{y^{2} - 25} \div \frac{2 x^{2} - 6 x}{3 y^{2} - 15 y} + \frac{3 - 1.5 y}{y + 5}$

Step 1

Simplify each term.

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Step 1.1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 9}{y^{2} - 25} \cdot \frac{3 y^{2} - 15 y}{2 x^{2} - 6 x} + \frac{3 - 1.5 y}{y + 5}$

Step 1.2

Simplify the numerator.

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Step 1.2.1

Rewrite $9$ as $3^{2}$ .

$\frac{x^{2} - 3^{2}}{y^{2} - 25} \cdot \frac{3 y^{2} - 15 y}{2 x^{2} - 6 x} + \frac{3 - 1.5 y}{y + 5}$

Step 1.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$\frac{(x + 3) (x - 3)}{y^{2} - 25} \cdot \frac{3 y^{2} - 15 y}{2 x^{2} - 6 x} + \frac{3 - 1.5 y}{y + 5}$

Step 1.3

Simplify the denominator.

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Step 1.3.1

Rewrite $25$ as $5^{2}$ .

$\frac{(x + 3) (x - 3)}{y^{2} - 5^{2}} \cdot \frac{3 y^{2} - 15 y}{2 x^{2} - 6 x} + \frac{3 - 1.5 y}{y + 5}$

Step 1.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = 5$ .

$\frac{(x + 3) (x - 3)}{(y + 5) (y - 5)} \cdot \frac{3 y^{2} - 15 y}{2 x^{2} - 6 x} + \frac{3 - 1.5 y}{y + 5}$

Step 1.4

Factor $3 y$ out of $3 y^{2} - 15 y$ .

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Step 1.4.1

Factor $3 y$ out of $3 y^{2}$ .

$\frac{(x + 3) (x - 3)}{(y + 5) (y - 5)} \cdot \frac{3 y (y) - 15 y}{2 x^{2} - 6 x} + \frac{3 - 1.5 y}{y + 5}$

Step 1.4.2

Factor $3 y$ out of $- 15 y$ .

$\frac{(x + 3) (x - 3)}{(y + 5) (y - 5)} \cdot \frac{3 y (y) + 3 y (- 5)}{2 x^{2} - 6 x} + \frac{3 - 1.5 y}{y + 5}$

Step 1.4.3

Factor $3 y$ out of $3 y (y) + 3 y (- 5)$ .

$\frac{(x + 3) (x - 3)}{(y + 5) (y - 5)} \cdot \frac{3 y (y - 5)}{2 x^{2} - 6 x} + \frac{3 - 1.5 y}{y + 5}$

Step 1.5

Factor $2 x$ out of $2 x^{2} - 6 x$ .

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Step 1.5.1

Factor $2 x$ out of $2 x^{2}$ .

$\frac{(x + 3) (x - 3)}{(y + 5) (y - 5)} \cdot \frac{3 y (y - 5)}{2 x (x) - 6 x} + \frac{3 - 1.5 y}{y + 5}$

Step 1.5.2

Factor $2 x$ out of $- 6 x$ .

$\frac{(x + 3) (x - 3)}{(y + 5) (y - 5)} \cdot \frac{3 y (y - 5)}{2 x (x) + 2 x (- 3)} + \frac{3 - 1.5 y}{y + 5}$

Step 1.5.3

Factor $2 x$ out of $2 x (x) + 2 x (- 3)$ .

$\frac{(x + 3) (x - 3)}{(y + 5) (y - 5)} \cdot \frac{3 y (y - 5)}{2 x (x - 3)} + \frac{3 - 1.5 y}{y + 5}$

Step 1.6

Cancel the common factor of $x - 3$ .

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Step 1.6.1

Factor $x - 3$ out of $(x + 3) (x - 3)$ .

$\frac{(x - 3) (x + 3)}{(y + 5) (y - 5)} \cdot \frac{3 y (y - 5)}{2 x (x - 3)} + \frac{3 - 1.5 y}{y + 5}$

Step 1.6.2

Factor $x - 3$ out of $2 x (x - 3)$ .

$\frac{(x - 3) (x + 3)}{(y + 5) (y - 5)} \cdot \frac{3 y (y - 5)}{(x - 3) (2 x)} + \frac{3 - 1.5 y}{y + 5}$

Step 1.6.3

Cancel the common factor.

$\frac{(x - 3) (x + 3)}{(y + 5) (y - 5)} \cdot \frac{3 y (y - 5)}{(x - 3) (2 x)} + \frac{3 - 1.5 y}{y + 5}$

Step 1.6.4

Rewrite the expression.

$\frac{x + 3}{(y + 5) (y - 5)} \cdot \frac{3 y (y - 5)}{2 x} + \frac{3 - 1.5 y}{y + 5}$

Step 1.7

Cancel the common factor of $y - 5$ .

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Step 1.7.1

Factor $y - 5$ out of $(y + 5) (y - 5)$ .

$\frac{x + 3}{(y - 5) (y + 5)} \cdot \frac{3 y (y - 5)}{2 x} + \frac{3 - 1.5 y}{y + 5}$

Step 1.7.2

Factor $y - 5$ out of $3 y (y - 5)$ .

$\frac{x + 3}{(y - 5) (y + 5)} \cdot \frac{(y - 5) (3 y)}{2 x} + \frac{3 - 1.5 y}{y + 5}$

Step 1.7.3

Cancel the common factor.

$\frac{x + 3}{(y - 5) (y + 5)} \cdot \frac{(y - 5) (3 y)}{2 x} + \frac{3 - 1.5 y}{y + 5}$

Step 1.7.4

Rewrite the expression.

$\frac{x + 3}{y + 5} \cdot \frac{3 y}{2 x} + \frac{3 - 1.5 y}{y + 5}$

Step 1.8

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

$\frac{(x + 3) (3 y)}{(y + 5) (2 x)} + \frac{3 - 1.5 y}{y + 5}$

Step 1.9

Move $3$ to the left of $x + 3$ .

$\frac{3 \cdot (x + 3) y}{(y + 5) \cdot 2 x} + \frac{3 - 1.5 y}{y + 5}$

Step 1.10

Move $2$ to the left of $y + 5$ .

$\frac{3 \cdot (x + 3) y}{2 \cdot (y + 5) x} + \frac{3 - 1.5 y}{y + 5}$

Step 1.11

Factor $1.5$ out of $3 - 1.5 y$ .

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Step 1.11.1

Factor $1.5$ out of $3$ .

$\frac{3 (x + 3) y}{2 (y + 5) x} + \frac{1.5 (2) - 1.5 y}{y + 5}$

Step 1.11.2

Factor $1.5$ out of $- 1.5 y$ .

$\frac{3 (x + 3) y}{2 (y + 5) x} + \frac{1.5 (2) + 1.5 (- y)}{y + 5}$

Step 1.11.3

Factor $1.5$ out of $1.5 (2) + 1.5 (- y)$ .

$\frac{3 (x + 3) y}{2 (y + 5) x} + \frac{1.5 (2 - y)}{y + 5}$

Step 2

To write $\frac{1.5 (2 - y)}{y + 5}$ as a fraction with a common denominator, multiply by $\frac{2 x}{2 x}$ .

$\frac{3 (x + 3) y}{2 (y + 5) x} + \frac{1.5 (2 - y)}{y + 5} \cdot \frac{2 x}{2 x}$

Step 3

Write each expression with a common denominator of $2 (y + 5) x$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{1.5 (2 - y)}{y + 5}$ by $\frac{2 x}{2 x}$ .

$\frac{3 (x + 3) y}{2 (y + 5) x} + \frac{1.5 (2 - y) (2 x)}{(y + 5) (2 x)}$

Step 3.2

Reorder the factors of $2 (y + 5) x$ .

$\frac{3 (x + 3) y}{2 x (y + 5)} + \frac{1.5 (2 - y) (2 x)}{(y + 5) (2 x)}$

Step 3.3

Reorder the factors of $(y + 5) (2 x)$ .

$\frac{3 (x + 3) y}{2 x (y + 5)} + \frac{1.5 (2 - y) (2 x)}{2 x (y + 5)}$

Step 4

Combine the numerators over the common denominator.

$\frac{3 (x + 3) y + 1.5 (2 - y) (2 x)}{2 x (y + 5)}$

Step 5

Simplify the numerator.

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Step 5.1

Factor $1.5$ out of $3 (x + 3) y + 1.5 (2 - y) \cdot 2 x$ .

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Step 5.1.1

Reorder the expression.

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Step 5.1.1.1

Move $x + 3$ .

$\frac{3 y (x + 3) + 1.5 (2 - y) \cdot 2 x}{2 x (y + 5)}$

Step 5.1.1.2

Reorder $2$ and $- y$ .

$\frac{3 y (x + 3) + 1.5 (- y + 2) \cdot 2 x}{2 x (y + 5)}$

Step 5.1.1.3

Move $- y + 2$ .

$\frac{3 y (x + 3) + 1.5 \cdot 2 x (- y + 2)}{2 x (y + 5)}$

Step 5.1.2

Factor $1.5$ out of $3 y (x + 3)$ .

$\frac{1.5 (2 y (x + 3)) + 1.5 \cdot 2 x (- y + 2)}{2 x (y + 5)}$

Step 5.1.3

Factor $1.5$ out of $1.5 \cdot 2 x (- y + 2)$ .

$\frac{1.5 (2 y (x + 3)) + 1.5 (2 x (- y + 2))}{2 x (y + 5)}$

Step 5.1.4

Factor $1.5$ out of $1.5 (2 y (x + 3)) + 1.5 (2 x (- y + 2))$ .

$\frac{1.5 (2 y (x + 3) + 2 x (- y + 2))}{2 x (y + 5)}$

Step 5.2

Factor $2$ out of $2 y (x + 3) + 2 x (- y + 2)$ .

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Step 5.2.1

Factor $2$ out of $2 y (x + 3)$ .

$\frac{1.5 (2 (y (x + 3)) + 2 x (- y + 2))}{2 x (y + 5)}$

Step 5.2.2

Factor $2$ out of $2 x (- y + 2)$ .

$\frac{1.5 (2 (y (x + 3)) + 2 (x (- y + 2)))}{2 x (y + 5)}$

Step 5.2.3

Factor $2$ out of $2 (y (x + 3)) + 2 (x (- y + 2))$ .

$\frac{1.5 (2 (y (x + 3) + x (- y + 2)))}{2 x (y + 5)}$

Step 5.3

Apply the distributive property.

$\frac{1.5 (2 (y x + y \cdot 3 + x (- y + 2)))}{2 x (y + 5)}$

Step 5.4

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

$\frac{1.5 (2 (y x + 3 \cdot y + x (- y + 2)))}{2 x (y + 5)}$

Step 5.5

Apply the distributive property.

$\frac{1.5 (2 (y x + 3 y + x (- y) + x \cdot 2))}{2 x (y + 5)}$

Step 5.6

Rewrite using the commutative property of multiplication.

$\frac{1.5 (2 (y x + 3 y - x y + x \cdot 2))}{2 x (y + 5)}$

Step 5.7

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

$\frac{1.5 (2 (y x + 3 y - x y + 2 \cdot x))}{2 x (y + 5)}$

Step 5.8

Subtract $x y$ from $y x$ .

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Step 5.8.1

Reorder $y$ and $x$ .

$\frac{1.5 (2 (3 y + x y - x y + 2 x))}{2 x (y + 5)}$

Step 5.8.2

Subtract $x y$ from $x y$ .

$\frac{1.5 (2 (3 y + 0 + 2 x))}{2 x (y + 5)}$

Step 5.9

Add $3 y$ and $0$ .

$\frac{1.5 \cdot 2 (3 y + 2 x)}{2 x (y + 5)}$

Step 5.10

Multiply $1.5$ by $2$ .

$\frac{3 (3 y + 2 x)}{2 x (y + 5)}$

Step 6

Cancel the common factor of $3$ and $2$ .

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Step 6.1

Factor $1$ out of $3 (3 y + 2 x)$ .

$\frac{1 (3 (3 y + 2 x))}{2 x (y + 5)}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $1$ out of $2 x (y + 5)$ .

$\frac{1 (3 (3 y + 2 x))}{1 (2 x (y + 5))}$

Step 6.2.2

Cancel the common factor.

$\frac{1 (3 (3 y + 2 x))}{1 (2 x (y + 5))}$

Step 6.2.3

Rewrite the expression.

$\frac{3 (3 y + 2 x)}{2 x (y + 5)}$