Question

In the following exercises, solve.

$\frac{m}{m+5}=\frac{50}{m^2-25}+6$

Short answer

The solution is$m=-3,2.$

Step-by-step solution

Step 1. Given Information

The given rational equation is$\frac{m}{m+5}=\frac{50}{m^2-25}+6$

Step 2. Least common Denominator(LCD)

The LCD of the equation is $(m+5)(m-5)$

Multiply the equation by LCD,

localid="1647797171568" $$\begin{gathered} (m+5)(m-5)\frac{m}{m+5}=\left[\frac{50}{m^2-25}+6\right](m+5)(m-5) \\ m(m-5)=50+6(m+5)(m-5) \\ {m}^2-5m=50+6(m^2-25) \\ {m}^2-5m=50+6m^2-25 \\ 5m^2+5m-25=0 \\ 5(m^2+m-5)=0 \end{gathered}$$

Step 3. Solution of the equation

On simplifying,

$$\begin{gathered} 5(m^2+m-5)=0 \\ 5(m^2+3m-2m-5)=0 \\ 5(m+3)(m-2)=0 \end{gathered}$$

Using the Zero product property,

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