Solve r-5/r-3-3/r-6

$$\frac { r - 5 } { r - 3 } - \frac { 3 } { r - 6 }$$

Evaluate

$$\frac{r^{2}-14r+39}{\left(r-6\right)\left(r-3\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of r-3 and r-6 is \left(r-6\right)\left(r-3\right). Multiply \frac{r-5}{r-3} times \frac{r-6}{r-6}. Multiply \frac{3}{r-6} times \frac{r-3}{r-3}.

$$\frac{\left(r-5\right)\left(r-6\right)}{\left(r-6\right)\left(r-3\right)}-\frac{3\left(r-3\right)}{\left(r-6\right)\left(r-3\right)}$$

Since \frac{\left(r-5\right)\left(r-6\right)}{\left(r-6\right)\left(r-3\right)} and \frac{3\left(r-3\right)}{\left(r-6\right)\left(r-3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(r-5\right)\left(r-6\right)-3\left(r-3\right)}{\left(r-6\right)\left(r-3\right)}$$

Do the multiplications in \left(r-5\right)\left(r-6\right)-3\left(r-3\right).

$$\frac{r^{2}-6r-5r+30-3r+9}{\left(r-6\right)\left(r-3\right)}$$

Combine like terms in r^{2}-6r-5r+30-3r+9.

$$\frac{r^{2}-14r+39}{\left(r-6\right)\left(r-3\right)}$$

Expand \left(r-6\right)\left(r-3\right).

$$\frac{r^{2}-14r+39}{r^{2}-9r+18}$$

Expand

$$\frac{r^{2}-14r+39}{\left(r-6\right)\left(r-3\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of r-3 and r-6 is \left(r-6\right)\left(r-3\right). Multiply \frac{r-5}{r-3} times \frac{r-6}{r-6}. Multiply \frac{3}{r-6} times \frac{r-3}{r-3}.

$$\frac{\left(r-5\right)\left(r-6\right)}{\left(r-6\right)\left(r-3\right)}-\frac{3\left(r-3\right)}{\left(r-6\right)\left(r-3\right)}$$

Since \frac{\left(r-5\right)\left(r-6\right)}{\left(r-6\right)\left(r-3\right)} and \frac{3\left(r-3\right)}{\left(r-6\right)\left(r-3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(r-5\right)\left(r-6\right)-3\left(r-3\right)}{\left(r-6\right)\left(r-3\right)}$$

Do the multiplications in \left(r-5\right)\left(r-6\right)-3\left(r-3\right).

$$\frac{r^{2}-6r-5r+30-3r+9}{\left(r-6\right)\left(r-3\right)}$$

Combine like terms in r^{2}-6r-5r+30-3r+9.

$$\frac{r^{2}-14r+39}{\left(r-6\right)\left(r-3\right)}$$

Expand \left(r-6\right)\left(r-3\right).

$$\frac{r^{2}-14r+39}{r^{2}-9r+18}$$