Simplify (r-2)/(r-1)-5/3

$\frac{r - 2}{r - 1} - \frac{5}{3}$

To write $\frac{r - 2}{r - 1}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{r - 2}{r - 1} \cdot \frac{3}{3} - \frac{5}{3}$

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

$\frac{r - 2}{r - 1} \cdot \frac{3}{3} - \frac{5}{3} \cdot \frac{r - 1}{r - 1}$

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

Multiply $\frac{r - 2}{r - 1}$ by $\frac{3}{3}$ .

$\frac{(r - 2) \cdot 3}{(r - 1) \cdot 3} - \frac{5}{3} \cdot \frac{r - 1}{r - 1}$

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

$\frac{(r - 2) \cdot 3}{(r - 1) \cdot 3} - \frac{5 (r - 1)}{3 (r - 1)}$

Reorder the factors of $(r - 1) \cdot 3$ .

$\frac{(r - 2) \cdot 3}{3 (r - 1)} - \frac{5 (r - 1)}{3 (r - 1)}$

Combine the numerators over the common denominator.

$\frac{(r - 2) \cdot 3 - 5 (r - 1)}{3 (r - 1)}$

Simplify the numerator.

Apply the distributive property.

$\frac{r \cdot 3 - 2 \cdot 3 - 5 (r - 1)}{3 (r - 1)}$

Move $3$ to the left of $r$ .

$\frac{3 \cdot r - 2 \cdot 3 - 5 (r - 1)}{3 (r - 1)}$

Multiply $- 2$ by $3$ .

$\frac{3 \cdot r - 6 - 5 (r - 1)}{3 (r - 1)}$

Apply the distributive property.

$\frac{3 r - 6 - 5 r - 5 \cdot - 1}{3 (r - 1)}$

Multiply $- 5$ by $- 1$ .

$\frac{3 r - 6 - 5 r + 5}{3 (r - 1)}$

Subtract $5 r$ from $3 r$ .

$\frac{- 2 r - 6 + 5}{3 (r - 1)}$

Add $- 6$ and $5$ .

$\frac{- 2 r - 1}{3 (r - 1)}$

Simplify with factoring out.

Factor $- 1$ out of $- 2 r$ .

$\frac{- (2 r) - 1}{3 (r - 1)}$

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

$\frac{- (2 r) - 1 (1)}{3 (r - 1)}$

Factor $- 1$ out of $- (2 r) - 1 (1)$ .

$\frac{- (2 r + 1)}{3 (r - 1)}$

Simplify the expression.

Rewrite $- (2 r + 1)$ as $- 1 (2 r + 1)$ .

$\frac{- 1 (2 r + 1)}{3 (r - 1)}$

Move the negative in front of the fraction.

$- \frac{2 r + 1}{3 (r - 1)}$