Question

In the following exercises, solve.

$\frac{1}{t+7}-\frac{5}{t-5}=1$

Short answer

The answer is$t=-5,-1.$

Step-by-step solution

Step 1. Given Information

The given rational expression is $\frac{1}{t+7}-\frac{5}{t-5}=1$

Step 2. Eliminating the fractions

The Least Common denominator is $(t+7)(t-5)$

Multiply the equation by $(t+7)(t-5)$:

$$\begin{gathered} (t+7)(t-5)\left\{\frac{1}{t+7}-\frac{5}{t-5}\right\}=1(t+7)(t-5) \\ (t-5)-5(t+7)=(t+7)(t-5) \\ t-5-5t-35=t^2-5t+7t-35 \\ -4t-40=t^2+2t-35 \\ {t}^2+6t+5=0 \end{gathered}$$

Step 3. Solving the Quadratic equation

The equation can be written as:

$$\begin{gathered} t^2+6t+5=0 \\ {t}^2+5t+t+5=0 \\ t(t+5)+1(t+5)=0 \\ (t+5)(t+1)=0 \end{gathered}$$

Using the Zero product property,

$$\begin{gathered} t+5=0;t+1=0 \\ t=-5;t=-1 \end{gathered}$$