Solve by Factoring 5/(r-9)=8/(r+5)

$\frac{5}{r - 9} = \frac{8}{r + 5}$

Step 1

Subtract $\frac{8}{r + 5}$ from both sides of the equation.

$\frac{5}{r - 9} - \frac{8}{r + 5} = 0$

Step 2

Simplify $\frac{5}{r - 9} - \frac{8}{r + 5}$ .

To write $\frac{5}{r - 9}$ as a fraction with a common denominator, multiply by $\frac{r + 5}{r + 5}$ .

$\frac{5}{r - 9} \cdot \frac{r + 5}{r + 5} - \frac{8}{r + 5} = 0$

To write $- \frac{8}{r + 5}$ as a fraction with a common denominator, multiply by $\frac{r - 9}{r - 9}$ .

$\frac{5}{r - 9} \cdot \frac{r + 5}{r + 5} - \frac{8}{r + 5} \cdot \frac{r - 9}{r - 9} = 0$

Write each expression with a common denominator of $(r - 9) (r + 5)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{5}{r - 9}$ by $\frac{r + 5}{r + 5}$ .

$\frac{5 (r + 5)}{(r - 9) (r + 5)} - \frac{8}{r + 5} \cdot \frac{r - 9}{r - 9} = 0$

Multiply $\frac{8}{r + 5}$ by $\frac{r - 9}{r - 9}$ .

$\frac{5 (r + 5)}{(r - 9) (r + 5)} - \frac{8 (r - 9)}{(r + 5) (r - 9)} = 0$

Reorder the factors of $(r - 9) (r + 5)$ .

$\frac{5 (r + 5)}{(r + 5) (r - 9)} - \frac{8 (r - 9)}{(r + 5) (r - 9)} = 0$

Combine the numerators over the common denominator.

$\frac{5 (r + 5) - 8 (r - 9)}{(r + 5) (r - 9)} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{5 r + 5 \cdot 5 - 8 (r - 9)}{(r + 5) (r - 9)} = 0$

Multiply $5$ by $5$ .

$\frac{5 r + 25 - 8 (r - 9)}{(r + 5) (r - 9)} = 0$

Apply the distributive property.

$\frac{5 r + 25 - 8 r - 8 \cdot - 9}{(r + 5) (r - 9)} = 0$

Multiply $- 8$ by $- 9$ .

$\frac{5 r + 25 - 8 r + 72}{(r + 5) (r - 9)} = 0$

Subtract $8 r$ from $5 r$ .

$\frac{- 3 r + 25 + 72}{(r + 5) (r - 9)} = 0$

Add $25$ and $72$ .

$\frac{- 3 r + 97}{(r + 5) (r - 9)} = 0$

Factor $- 1$ out of $- 3 r$ .

$\frac{- (3 r) + 97}{(r + 5) (r - 9)} = 0$

Rewrite $97$ as $- 1 (- 97)$ .

$\frac{- (3 r) - 1 (- 97)}{(r + 5) (r - 9)} = 0$

Factor $- 1$ out of $- (3 r) - 1 (- 97)$ .

$\frac{- (3 r - 97)}{(r + 5) (r - 9)} = 0$

Rewrite $- (3 r - 97)$ as $- 1 (3 r - 97)$ .

$\frac{- 1 (3 r - 97)}{(r + 5) (r - 9)} = 0$

Move the negative in front of the fraction.

$- \frac{3 r - 97}{(r + 5) (r - 9)} = 0$

Step 3

Set the numerator equal to zero.

$3 r - 97 = 0$

Step 4

Solve the equation for $r$ .

Add $97$ to both sides of the equation.

$3 r = 97$

Divide each term in $3 r = 97$ by $3$ and simplify.

Divide each term in $3 r = 97$ by $3$ .

$\frac{3 r}{3} = \frac{97}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 r}{3} = \frac{97}{3}$

Divide $r$ by $1$ .

$r = \frac{97}{3}$

Solve by Factoring 5/(r-9)=8/(r+5)

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