Algebra Examples

$\frac{x + 3}{x - 6} - \frac{7}{x - 3}$

Step 1

To write $\frac{x + 3}{x - 6}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{x + 3}{x - 6} \cdot \frac{x - 3}{x - 3} - \frac{7}{x - 3}$

Step 2

To write $- \frac{7}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{x - 6}{x - 6}$ .

$\frac{x + 3}{x - 6} \cdot \frac{x - 3}{x - 3} - \frac{7}{x - 3} \cdot \frac{x - 6}{x - 6}$

Step 3

Write each expression with a common denominator of $(x - 6) (x - 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{x + 3}{x - 6}$ by $\frac{x - 3}{x - 3}$ .

$\frac{(x + 3) (x - 3)}{(x - 6) (x - 3)} - \frac{7}{x - 3} \cdot \frac{x - 6}{x - 6}$

Step 3.2

Multiply $\frac{7}{x - 3}$ by $\frac{x - 6}{x - 6}$ .

$\frac{(x + 3) (x - 3)}{(x - 6) (x - 3)} - \frac{7 (x - 6)}{(x - 3) (x - 6)}$

Step 3.3

Reorder the factors of $(x - 6) (x - 3)$ .

$\frac{(x + 3) (x - 3)}{(x - 3) (x - 6)} - \frac{7 (x - 6)}{(x - 3) (x - 6)}$

Step 4

Combine the numerators over the common denominator.

$\frac{(x + 3) (x - 3) - 7 (x - 6)}{(x - 3) (x - 6)}$

Step 5

Simplify the numerator.

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Step 5.1

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

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Step 5.1.1

Apply the distributive property.

$\frac{x (x - 3) + 3 (x - 3) - 7 (x - 6)}{(x - 3) (x - 6)}$

Step 5.1.2

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 3 + 3 (x - 3) - 7 (x - 6)}{(x - 3) (x - 6)}$

Step 5.1.3

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 3 + 3 x + 3 \cdot - 3 - 7 (x - 6)}{(x - 3) (x - 6)}$

Step 5.2

Combine the opposite terms in $x \cdot x + x \cdot - 3 + 3 x + 3 \cdot - 3$ .

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Step 5.2.1

Reorder the factors in the terms $x \cdot - 3$ and $3 x$ .

$\frac{x \cdot x - 3 x + 3 x + 3 \cdot - 3 - 7 (x - 6)}{(x - 3) (x - 6)}$

Step 5.2.2

Add $- 3 x$ and $3 x$ .

$\frac{x \cdot x + 0 + 3 \cdot - 3 - 7 (x - 6)}{(x - 3) (x - 6)}$

Step 5.2.3

Add $x \cdot x$ and $0$ .

$\frac{x \cdot x + 3 \cdot - 3 - 7 (x - 6)}{(x - 3) (x - 6)}$

Step 5.3

Simplify each term.

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Step 5.3.1

Multiply $x$ by $x$ .

$\frac{x^{2} + 3 \cdot - 3 - 7 (x - 6)}{(x - 3) (x - 6)}$

Step 5.3.2

Multiply $3$ by $- 3$ .

$\frac{x^{2} - 9 - 7 (x - 6)}{(x - 3) (x - 6)}$

Step 5.4

Apply the distributive property.

$\frac{x^{2} - 9 - 7 x - 7 \cdot - 6}{(x - 3) (x - 6)}$

Step 5.5

Multiply $- 7$ by $- 6$ .

$\frac{x^{2} - 9 - 7 x + 42}{(x - 3) (x - 6)}$

Step 5.6

Add $- 9$ and $42$ .

$\frac{x^{2} - 7 x + 33}{(x - 3) (x - 6)}$