Algebra Examples

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

Step 1

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

$\frac{x - y}{x y} \cdot \frac{z}{z} - \frac{x - z}{x z}$

Step 2

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

$\frac{x - y}{x y} \cdot \frac{z}{z} - \frac{x - z}{x z} \cdot \frac{y}{y}$

Step 3

Write each expression with a common denominator of $x y z$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{x - y}{x y}$ by $\frac{z}{z}$ .

$\frac{(x - y) z}{x y z} - \frac{x - z}{x z} \cdot \frac{y}{y}$

Step 3.2

Multiply $\frac{x - z}{x z}$ by $\frac{y}{y}$ .

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

Step 3.3

Reorder the factors of $x z y$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Multiply $- - z$ .

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Step 5.3.1

Multiply $- 1$ by $- 1$ .

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

Step 5.3.2

Multiply $z$ by $1$ .

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

Step 5.4

Apply the distributive property.

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

Step 5.5

Add $- y z$ and $z y$ .

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Step 5.5.1

Reorder $z$ and $y$ .

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

Step 5.5.2

Add $- y z$ and $y z$ .

$\frac{x z - x y + 0}{x y z}$

Step 5.6

Add $x z - x y$ and $0$ .

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

Step 5.7

Rewrite $x z - x y$ in a factored form.

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Step 5.7.1

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

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Step 5.7.1.1

Factor $x$ out of $x z$ .

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

Step 5.7.1.2

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

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

Step 5.7.1.3

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

$\frac{x (z - 1 y)}{x y z}$

Step 5.7.2

Rewrite $- 1 y$ as $- y$ .

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

Step 6

Cancel the common factor of $x$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

$\frac{z - y}{y z}$