Simplify 7/(x-3)+(3x+1)/(x-5)

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

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 $1$ .

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

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

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

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

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.

Apply the distributive property.

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

Multiply $7$ by $- 5$ .

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 3$ by $3$ .

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

Multiply $x$ by $1$ .

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

Multiply $- 3$ by $1$ .

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

Add $- 9 x$ and $x$ .

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

Subtract $8 x$ from $7 x$ .

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

Subtract $3$ from $- 35$ .

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

Reorder terms.

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

Simplify 7/(x-3)+(3x+1)/(x-5)

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