Simplify 3/(n-5)+6/(3n-8)

$\frac{3}{n - 5} + \frac{6}{3 n - 8}$

Step 1

To write $\frac{3}{n - 5}$ as a fraction with a common denominator, multiply by $\frac{3 n - 8}{3 n - 8}$ .

$\frac{3}{n - 5} \cdot \frac{3 n - 8}{3 n - 8} + \frac{6}{3 n - 8}$

Step 2

To write $\frac{6}{3 n - 8}$ as a fraction with a common denominator, multiply by $\frac{n - 5}{n - 5}$ .

$\frac{3}{n - 5} \cdot \frac{3 n - 8}{3 n - 8} + \frac{6}{3 n - 8} \cdot \frac{n - 5}{n - 5}$

Step 3

Write each expression with a common denominator of $(n - 5) (3 n - 8)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3}{n - 5}$ by $\frac{3 n - 8}{3 n - 8}$ .

$\frac{3 (3 n - 8)}{(n - 5) (3 n - 8)} + \frac{6}{3 n - 8} \cdot \frac{n - 5}{n - 5}$

Multiply $\frac{6}{3 n - 8}$ by $\frac{n - 5}{n - 5}$ .

$\frac{3 (3 n - 8)}{(n - 5) (3 n - 8)} + \frac{6 (n - 5)}{(3 n - 8) (n - 5)}$

Reorder the factors of $(n - 5) (3 n - 8)$ .

$\frac{3 (3 n - 8)}{(3 n - 8) (n - 5)} + \frac{6 (n - 5)}{(3 n - 8) (n - 5)}$

Step 4

Combine the numerators over the common denominator.

$\frac{3 (3 n - 8) + 6 (n - 5)}{(3 n - 8) (n - 5)}$

Step 5

Simplify the numerator.

Factor $3$ out of $3 (3 n - 8) + 6 (n - 5)$ .

Factor $3$ out of $6 (n - 5)$ .

$\frac{3 (3 n - 8) + 3 (2 (n - 5))}{(3 n - 8) (n - 5)}$

Factor $3$ out of $3 (3 n - 8) + 3 (2 (n - 5))$ .

$\frac{3 (3 n - 8 + 2 (n - 5))}{(3 n - 8) (n - 5)}$

Apply the distributive property.

$\frac{3 (3 n - 8 + 2 n + 2 \cdot - 5)}{(3 n - 8) (n - 5)}$

Multiply $2$ by $- 5$ .

$\frac{3 (3 n - 8 + 2 n - 10)}{(3 n - 8) (n - 5)}$

Add $3 n$ and $2 n$ .

$\frac{3 (5 n - 8 - 10)}{(3 n - 8) (n - 5)}$

Subtract $10$ from $- 8$ .

$\frac{3 (5 n - 18)}{(3 n - 8) (n - 5)}$