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

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

Step 1

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

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

Step 2

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

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

Step 3

Write each expression with a common denominator of $(n - 3) (5 n - 6)$ , by multiplying each by an appropriate factor of $1$ .

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

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

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

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

Reorder the factors of $(n - 3) (5 n - 6)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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

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

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

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

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

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

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

Apply the distributive property.

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

Multiply $5$ by $8$ .

$\frac{n (40 n + 8 \cdot - 6 + 3 (n - 3))}{(5 n - 6) (n - 3)}$

Multiply $8$ by $- 6$ .

$\frac{n (40 n - 48 + 3 (n - 3))}{(5 n - 6) (n - 3)}$

Apply the distributive property.

$\frac{n (40 n - 48 + 3 n + 3 \cdot - 3)}{(5 n - 6) (n - 3)}$

Multiply $3$ by $- 3$ .

$\frac{n (40 n - 48 + 3 n - 9)}{(5 n - 6) (n - 3)}$

Add $40 n$ and $3 n$ .

$\frac{n (43 n - 48 - 9)}{(5 n - 6) (n - 3)}$

Subtract $9$ from $- 48$ .

$\frac{n (43 n - 57)}{(5 n - 6) (n - 3)}$