Algebra Examples

$\frac{\frac{3 x + y}{x - y} - 3}{1 - \frac{x - 3 y}{x + y}}$

Step 1

Multiply the numerator and denominator of the fraction by $(x - y) (x + y)$ .

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Step 1.1

Multiply $\frac{\frac{3 x + y}{x - y} - 3}{1 - \frac{x - 3 y}{x + y}}$ by $\frac{(x - y) (x + y)}{(x - y) (x + y)}$ .

$\frac{(x - y) (x + y)}{(x - y) (x + y)} \cdot \frac{\frac{3 x + y}{x - y} - 3}{1 - \frac{x - 3 y}{x + y}}$

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

$\frac{(x - y) (x + y) \frac{3 x + y}{x - y} + (x - y) (x + y) \cdot - 3}{(x - y) (x + y) \cdot 1 + (x - y) (x + y) (- \frac{x - 3 y}{x + y})}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x - y$ .

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Step 3.1.1

Cancel the common factor.

$\frac{(x - y) (x + y) \frac{3 x + y}{x - y} + (x - y) (x + y) \cdot - 3}{(x - y) (x + y) \cdot 1 + (x - y) (x + y) (- \frac{x - 3 y}{x + y})}$

Step 3.1.2

Rewrite the expression.

$\frac{(x + y) (3 x + y) + (x - y) (x + y) \cdot - 3}{(x - y) (x + y) \cdot 1 + (x - y) (x + y) (- \frac{x - 3 y}{x + y})}$

Step 3.2

Cancel the common factor of $x + y$ .

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Step 3.2.1

Move the leading negative in $- \frac{x - 3 y}{x + y}$ into the numerator.

$\frac{(x + y) (3 x + y) + (x - y) (x + y) \cdot - 3}{(x - y) (x + y) \cdot 1 + (x - y) (x + y) \frac{- (x - 3 y)}{x + y}}$

Step 3.2.2

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

$\frac{(x + y) (3 x + y) + (x - y) (x + y) \cdot - 3}{(x - y) (x + y) \cdot 1 + (x + y) (x - y) \frac{- (x - 3 y)}{x + y}}$

Step 3.2.3

Cancel the common factor.

$\frac{(x + y) (3 x + y) + (x - y) (x + y) \cdot - 3}{(x - y) (x + y) \cdot 1 + (x + y) (x - y) \frac{- (x - 3 y)}{x + y}}$

Step 3.2.4

Rewrite the expression.

$\frac{(x + y) (3 x + y) + (x - y) (x + y) \cdot - 3}{(x - y) (x + y) \cdot 1 + (x - y) (- (x - 3 y))}$

Step 4

Simplify the numerator.

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Step 4.1

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

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Step 4.1.1

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

$\frac{(x + y) (3 x + y) + (x + y) ((x - y) \cdot - 3)}{(x - y) (x + y) \cdot 1 + (x - y) (- (x - 3 y))}$

Step 4.1.2

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

$\frac{(x + y) (3 x + y + (x - y) \cdot - 3)}{(x - y) (x + y) \cdot 1 + (x - y) (- (x - 3 y))}$

Step 4.2

Apply the distributive property.

$\frac{(x + y) (3 x + y + x \cdot - 3 - y \cdot - 3)}{(x - y) (x + y) \cdot 1 + (x - y) (- (x - 3 y))}$

Step 4.3

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

$\frac{(x + y) (3 x + y - 3 \cdot x - y \cdot - 3)}{(x - y) (x + y) \cdot 1 + (x - y) (- (x - 3 y))}$

Step 4.4

Multiply $- 3$ by $- 1$ .

$\frac{(x + y) (3 x + y - 3 \cdot x + 3 y)}{(x - y) (x + y) \cdot 1 + (x - y) (- (x - 3 y))}$

Step 4.5

Subtract $3 x$ from $3 x$ .

$\frac{(x + y) (y + 0 + 3 y)}{(x - y) (x + y) \cdot 1 + (x - y) (- (x - 3 y))}$

Step 4.6

Add $y$ and $0$ .

$\frac{(x + y) (y + 3 y)}{(x - y) (x + y) \cdot 1 + (x - y) (- (x - 3 y))}$

Step 4.7

Add $y$ and $3 y$ .

$\frac{(x + y) \cdot 4 y}{(x - y) (x + y) \cdot 1 + (x - y) (- (x - 3 y))}$

Step 5

Simplify the denominator.

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Step 5.1

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

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Step 5.1.1

Factor $x - y$ out of $(x - y) (x + y) \cdot 1$ .

$\frac{(x + y) \cdot 4 y}{(x - y) ((x + y) \cdot 1) + (x - y) (- (x - 3 y))}$

Step 5.1.2

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

$\frac{(x + y) \cdot 4 y}{(x - y) ((x + y) \cdot 1 - (x - 3 y))}$

Step 5.2

Multiply $x + y$ by $1$ .

$\frac{(x + y) \cdot 4 y}{(x - y) (x + y - (x - 3 y))}$

Step 5.3

Apply the distributive property.

$\frac{(x + y) \cdot 4 y}{(x - y) (x + y - x - (- 3 y))}$

Step 5.4

Multiply $- 3$ by $- 1$ .

$\frac{(x + y) \cdot 4 y}{(x - y) (x + y - x + 3 y)}$

Step 5.5

Subtract $x$ from $x$ .

$\frac{(x + y) \cdot 4 y}{(x - y) (y + 0 + 3 y)}$

Step 5.6

Add $y$ and $0$ .

$\frac{(x + y) \cdot 4 y}{(x - y) (y + 3 y)}$

Step 5.7

Add $y$ and $3 y$ .

$\frac{(x + y) \cdot 4 y}{(x - y) \cdot 4 y}$

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Cancel the common factor of $4$ .

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Step 6.1.1

Cancel the common factor.

$\frac{(x + y) \cdot 4 y}{(x - y) \cdot 4 y}$

Step 6.1.2

Rewrite the expression.

$\frac{(x + y) y}{(x - y) y}$

Step 6.2

Cancel the common factor of $y$ .

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Step 6.2.1

Cancel the common factor.

$\frac{(x + y) y}{(x - y) y}$

Step 6.2.2

Rewrite the expression.

$\frac{x + y}{x - y}$