Question

In the following exercises, solve.

$\frac{2}{s+7}-\frac{3}{s-3}=1$

Short answer

The answer is$s=-2,-3.$

Step-by-step solution

Step 1. Given information

The given rational expression is

$\frac{2}{s+7}-\frac{3}{s-3}=1$

Step 2. Eliminating the fractions 

The Least Common denominator is $(s+7)(s-3)$

Multiply the equation by LCD

$$\begin{gathered} (s+7)(s-3)\{\frac{2}{s+7}-\frac{3}{s-3}\}=1(s+7)(s-3) \\ 2(s-3)-3(s+7)=s^2-3s+7s-21 \\ 2s-6-3s-21=s^2+4s-21 \\ -s-27=s^2+4s-21 \\ {s}^2+5s+6=0 \end{gathered}$$

Step 3. Solving the Quadratic equation 

The equation can be written as:

$s^2+5s+6=0$

$$\begin{gathered} s^2+2s+3s+6=0 \\ s(s+2)+3(s+2)=0 \\ (s+3)(s+2)=0 \end{gathered}$$

Using the Zero product property,

$$\begin{gathered} s+3=0;s+2=0 \\ s=-3,s=-2 \end{gathered}$$