Algebra Examples

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

Step 1

To write $\frac{x - y}{2 x}$ as a fraction with a common denominator, multiply by $\frac{3 - x}{3 - x}$ .

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

Step 2

To write $\frac{5 - y}{3 - x}$ as a fraction with a common denominator, multiply by $\frac{2 x}{2 x}$ .

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

Step 3

Write each expression with a common denominator of $2 x (3 - x)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Reorder the factors of $(3 - x) (2 x)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Expand $(x - y) (3 - x)$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Simplify each term.

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Step 5.2.1

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

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

Step 5.2.2

Rewrite using the commutative property of multiplication.

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

Step 5.2.3

Multiply $x$ by $x$ by adding the exponents.

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Step 5.2.3.1

Move $x$ .

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

Step 5.2.3.2

Multiply $x$ by $x$ .

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

Step 5.2.4

Multiply $3$ by $- 1$ .

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

Step 5.2.5

Rewrite using the commutative property of multiplication.

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

Step 5.2.6

Multiply $- 1$ by $- 1$ .

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

Step 5.2.7

Multiply $y$ by $1$ .

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

Step 5.3

Apply the distributive property.

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

Step 5.4

Multiply $5$ by $2$ .

$\frac{3 x - x^{2} - 3 y + y x + (10 - y \cdot 2) x}{2 x (3 - x)}$

Step 5.5

Multiply $2$ by $- 1$ .

$\frac{3 x - x^{2} - 3 y + y x + (10 - 2 y) x}{2 x (3 - x)}$

Step 5.6

Apply the distributive property.

$\frac{3 x - x^{2} - 3 y + y x + 10 x - 2 y x}{2 x (3 - x)}$

Step 5.7

Add $3 x$ and $10 x$ .

$\frac{- x^{2} - 3 y + y x + 13 x - 2 y x}{2 x (3 - x)}$

Step 5.8

Subtract $2 y x$ from $y x$ .

$\frac{- x^{2} - 3 y - y x + 13 x}{2 x (3 - x)}$

Step 6

Simplify with factoring out.

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Step 6.1

Factor $- 1$ out of $- x^{2}$ .

$\frac{- (x^{2}) - 3 y - y x + 13 x}{2 x (3 - x)}$

Step 6.2

Factor $- 1$ out of $- 3 y$ .

$\frac{- (x^{2}) - (3 y) - y x + 13 x}{2 x (3 - x)}$

Step 6.3

Factor $- 1$ out of $- (x^{2}) - (3 y)$ .

$\frac{- (x^{2} + 3 y) - y x + 13 x}{2 x (3 - x)}$

Step 6.4

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

$\frac{- (x^{2} + 3 y) - (y x) + 13 x}{2 x (3 - x)}$

Step 6.5

Factor $- 1$ out of $- (x^{2} + 3 y) - (y x)$ .

$\frac{- (x^{2} + 3 y + y x) + 13 x}{2 x (3 - x)}$

Step 6.6

Factor $- 1$ out of $13 x$ .

$\frac{- (x^{2} + 3 y + y x) - (- 13 x)}{2 x (3 - x)}$

Step 6.7

Factor $- 1$ out of $- (x^{2} + 3 y + y x) - (- 13 x)$ .

$\frac{- (x^{2} + 3 y + y x - 13 x)}{2 x (3 - x)}$

Step 6.8

Simplify the expression.

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Step 6.8.1

Rewrite $- (x^{2} + 3 y + y x - 13 x)$ as $- 1 (x^{2} + 3 y + y x - 13 x)$ .

$\frac{- 1 (x^{2} + 3 y + y x - 13 x)}{2 x (3 - x)}$

Step 6.8.2

Move the negative in front of the fraction.

$- \frac{x^{2} + 3 y + y x - 13 x}{2 x (3 - x)}$