Pre-Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Write each expression with a common denominator of $(x - 7) (x - 6)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Reorder the factors of $(x - 7) (x - 6)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

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

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Step 5.2.1

Move $x$ .

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

Step 5.2.2

Multiply $x$ by $x$ .

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

Step 5.3

Multiply $- 6$ by $7$ .

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

Step 5.4

Expand $(7 x + 3) (x - 7)$ using the FOIL Method.

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Step 5.4.1

Apply the distributive property.

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

Step 5.4.2

Apply the distributive property.

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

Step 5.4.3

Apply the distributive property.

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

Step 5.5

Simplify and combine like terms.

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Step 5.5.1

Simplify each term.

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Step 5.5.1.1

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

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Step 5.5.1.1.1

Move $x$ .

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

Step 5.5.1.1.2

Multiply $x$ by $x$ .

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

Step 5.5.1.2

Multiply $- 7$ by $7$ .

$\frac{7 x^{2} - 42 x + 7 x^{2} - 49 x + 3 x + 3 \cdot - 7}{(x - 6) (x - 7)}$

Step 5.5.1.3

Multiply $3$ by $- 7$ .

$\frac{7 x^{2} - 42 x + 7 x^{2} - 49 x + 3 x - 21}{(x - 6) (x - 7)}$

Step 5.5.2

Add $- 49 x$ and $3 x$ .

$\frac{7 x^{2} - 42 x + 7 x^{2} - 46 x - 21}{(x - 6) (x - 7)}$

Step 5.6

Add $7 x^{2}$ and $7 x^{2}$ .

$\frac{14 x^{2} - 42 x - 46 x - 21}{(x - 6) (x - 7)}$

Step 5.7

Subtract $46 x$ from $- 42 x$ .

$\frac{14 x^{2} - 88 x - 21}{(x - 6) (x - 7)}$