Simplify (t+2)/(3t-9)+(3-2t)/(5t-15)

$\frac{t + 2}{3 t - 9} + \frac{3 - 2 t}{5 t - 15}$

Step 1

Simplify each term.

Factor $3$ out of $3 t - 9$ .

Factor $3$ out of $3 t$ .

$\frac{t + 2}{3 (t) - 9} + \frac{3 - 2 t}{5 t - 15}$

Factor $3$ out of $- 9$ .

$\frac{t + 2}{3 t + 3 \cdot - 3} + \frac{3 - 2 t}{5 t - 15}$

Factor $3$ out of $3 t + 3 \cdot - 3$ .

$\frac{t + 2}{3 (t - 3)} + \frac{3 - 2 t}{5 t - 15}$

Factor $5$ out of $5 t - 15$ .

Factor $5$ out of $5 t$ .

$\frac{t + 2}{3 (t - 3)} + \frac{3 - 2 t}{5 (t) - 15}$

Factor $5$ out of $- 15$ .

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

Factor $5$ out of $5 t + 5 \cdot - 3$ .

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

Step 2

To write $\frac{t + 2}{3 (t - 3)}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Step 3

To write $\frac{3 - 2 t}{5 (t - 3)}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 4

Write each expression with a common denominator of $15 (t - 3)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{t + 2}{3 (t - 3)}$ by $\frac{5}{5}$ .

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

Multiply $5$ by $3$ .

$\frac{(t + 2) \cdot 5}{15 (t - 3)} + \frac{3 - 2 t}{5 (t - 3)} \cdot \frac{3}{3}$

Multiply $\frac{3 - 2 t}{5 (t - 3)}$ by $\frac{3}{3}$ .

$\frac{(t + 2) \cdot 5}{15 (t - 3)} + \frac{(3 - 2 t) \cdot 3}{5 (t - 3) \cdot 3}$

Multiply $3$ by $5$ .

$\frac{(t + 2) \cdot 5}{15 (t - 3)} + \frac{(3 - 2 t) \cdot 3}{15 (t - 3)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(t + 2) \cdot 5 + (3 - 2 t) \cdot 3}{15 (t - 3)}$

Step 6

Simplify the numerator.

Apply the distributive property.

$\frac{t \cdot 5 + 2 \cdot 5 + (3 - 2 t) \cdot 3}{15 (t - 3)}$

Move $5$ to the left of $t$ .

$\frac{5 \cdot t + 2 \cdot 5 + (3 - 2 t) \cdot 3}{15 (t - 3)}$

Multiply $2$ by $5$ .

$\frac{5 \cdot t + 10 + (3 - 2 t) \cdot 3}{15 (t - 3)}$

Apply the distributive property.

$\frac{5 t + 10 + 3 \cdot 3 - 2 t \cdot 3}{15 (t - 3)}$

Multiply $3$ by $3$ .

$\frac{5 t + 10 + 9 - 2 t \cdot 3}{15 (t - 3)}$

Multiply $3$ by $- 2$ .

$\frac{5 t + 10 + 9 - 6 t}{15 (t - 3)}$

Subtract $6 t$ from $5 t$ .

$\frac{- t + 10 + 9}{15 (t - 3)}$

Add $10$ and $9$ .

$\frac{- t + 19}{15 (t - 3)}$

Step 7

Simplify with factoring out.

Factor $- 1$ out of $- t$ .

$\frac{- (t) + 19}{15 (t - 3)}$

Rewrite $19$ as $- 1 (- 19)$ .

$\frac{- (t) - 1 (- 19)}{15 (t - 3)}$

Factor $- 1$ out of $- (t) - 1 (- 19)$ .

$\frac{- (t - 19)}{15 (t - 3)}$

Simplify the expression.

Rewrite $- (t - 19)$ as $- 1 (t - 19)$ .

$\frac{- 1 (t - 19)}{15 (t - 3)}$

Move the negative in front of the fraction.

$- \frac{t - 19}{15 (t - 3)}$

Simplify (t+2)/(3t-9)+(3-2t)/(5t-15)

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