Pre-Algebra Examples

$\frac{7}{x - 3} + \frac{3 x + 1}{x - 5}$

Step 1

To write $\frac{7}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

$\frac{7}{x - 3} \cdot \frac{x - 5}{x - 5} + \frac{3 x + 1}{x - 5}$

Step 2

To write $\frac{3 x + 1}{x - 5}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{7}{x - 3} \cdot \frac{x - 5}{x - 5} + \frac{3 x + 1}{x - 5} \cdot \frac{x - 3}{x - 3}$

Step 3

Write each expression with a common denominator of $(x - 3) (x - 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{7}{x - 3}$ by $\frac{x - 5}{x - 5}$ .

$\frac{7 (x - 5)}{(x - 3) (x - 5)} + \frac{3 x + 1}{x - 5} \cdot \frac{x - 3}{x - 3}$

Step 3.2

Multiply $\frac{3 x + 1}{x - 5}$ by $\frac{x - 3}{x - 3}$ .

$\frac{7 (x - 5)}{(x - 3) (x - 5)} + \frac{(3 x + 1) (x - 3)}{(x - 5) (x - 3)}$

Step 3.3

Reorder the factors of $(x - 5) (x - 3)$ .

$\frac{7 (x - 5)}{(x - 3) (x - 5)} + \frac{(3 x + 1) (x - 3)}{(x - 3) (x - 5)}$

Step 4

Combine the numerators over the common denominator.

$\frac{7 (x - 5) + (3 x + 1) (x - 3)}{(x - 3) (x - 5)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{7 x + 7 \cdot - 5 + (3 x + 1) (x - 3)}{(x - 3) (x - 5)}$

Step 5.2

Multiply $7$ by $- 5$ .

$\frac{7 x - 35 + (3 x + 1) (x - 3)}{(x - 3) (x - 5)}$

Step 5.3

Expand $(3 x + 1) (x - 3)$ using the FOIL Method.

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Step 5.3.1

Apply the distributive property.

$\frac{7 x - 35 + 3 x (x - 3) + 1 (x - 3)}{(x - 3) (x - 5)}$

Step 5.3.2

Apply the distributive property.

$\frac{7 x - 35 + 3 x \cdot x + 3 x \cdot - 3 + 1 (x - 3)}{(x - 3) (x - 5)}$

Step 5.3.3

Apply the distributive property.

$\frac{7 x - 35 + 3 x \cdot x + 3 x \cdot - 3 + 1 x + 1 \cdot - 3}{(x - 3) (x - 5)}$

Step 5.4

Simplify and combine like terms.

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Step 5.4.1

Simplify each term.

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Step 5.4.1.1

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

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Step 5.4.1.1.1

Move $x$ .

$\frac{7 x - 35 + 3 (x \cdot x) + 3 x \cdot - 3 + 1 x + 1 \cdot - 3}{(x - 3) (x - 5)}$

Step 5.4.1.1.2

Multiply $x$ by $x$ .

$\frac{7 x - 35 + 3 x^{2} + 3 x \cdot - 3 + 1 x + 1 \cdot - 3}{(x - 3) (x - 5)}$

Step 5.4.1.2

Multiply $- 3$ by $3$ .

$\frac{7 x - 35 + 3 x^{2} - 9 x + 1 x + 1 \cdot - 3}{(x - 3) (x - 5)}$

Step 5.4.1.3

Multiply $x$ by $1$ .

$\frac{7 x - 35 + 3 x^{2} - 9 x + x + 1 \cdot - 3}{(x - 3) (x - 5)}$

Step 5.4.1.4

Multiply $- 3$ by $1$ .

$\frac{7 x - 35 + 3 x^{2} - 9 x + x - 3}{(x - 3) (x - 5)}$

Step 5.4.2

Add $- 9 x$ and $x$ .

$\frac{7 x - 35 + 3 x^{2} - 8 x - 3}{(x - 3) (x - 5)}$

Step 5.5

Subtract $8 x$ from $7 x$ .

$\frac{- x - 35 + 3 x^{2} - 3}{(x - 3) (x - 5)}$

Step 5.6

Subtract $3$ from $- 35$ .

$\frac{- x + 3 x^{2} - 38}{(x - 3) (x - 5)}$

Step 5.7

Reorder terms.

$\frac{3 x^{2} - x - 38}{(x - 3) (x - 5)}$