Question

Solve the equation $\frac{2a-5}{a-9}+\frac{a}{a+9}=\frac{-6}{a^2-81}$. Check your solution.

Short answer

The solution for the equation is$a=\left\{3,-\frac{13}{3}\right\}$.

Step-by-step solution

Step 1. Write down the given expression.

The given equation is $\frac{2a-5}{a-9}+\frac{a}{a+9}=\frac{-6}{a^2-81}$.

Step 2. Simplify the equation.

The given equation $\frac{2a-5}{a-9}+\frac{a}{a+9}=\frac{-6}{a^2-81}$is solved as:

$\begin{array}{l}\frac{2a-5}{a-9}+\frac{a}{a+9}=\frac{-6}{a^2-81}\mathrm{....}\left(\text{Given}\right)\\ \frac{2a-5}{a-9}+\frac{a}{a+9}=\frac{-6}{\left(a-9\right)\left(a+9\right)}\\ \left(2a-5\right)\left(a+9\right)+a\left(a-9\right)=-6\\ 2a^2+13a-45+a^2-9a=-6\\ 3a^2+4a-39=0\end{array}$

Further simplifying,

$\begin{array}{l}3a^2-9a+13a-39=\mathrm{0....}\left(\text{Factoring}\right)\\ 3a\left(a-3\right)+13\left(a-3\right)=0\\ \left(a-3\right)\left(3a+13\right)=0\\ a-3=0\text{or}3a+13=0\\ a=3\text{or}a=-\frac{13}{3}\\ a=\left\{3,-\frac{13}{3}\right\}\end{array}$

Step 3. Verify the result for obtained value.

Plugging $a=3$in given equation and verify the result.

$\begin{array}{l}\frac{2a-5}{a-9}+\frac{a}{a+9}=\frac{-6}{a^2-81}\mathrm{....}\left(\text{Given}\right)\\ \frac{2\left(3\right)-5}{\left(3\right)-9}+\frac{\left(3\right)}{\left(3\right)+9}=\frac{-6}{{\left(3\right)}^2-81}\mathrm{....}\left(\text{Plugging}a=3\right)\\ -\frac{1}{6}+\frac{1}{4}=\frac{-6}{-72}\\ \frac{-2+3}{12}=\frac{1}{12}\\ \frac{1}{12}=\frac{1}{12}\mathrm{....}\left(\text{True}\right)\end{array}$

Hence the solution is verified for $a=3$.

Step 4. Conclusion.

The solution for the given equation is $a=\left\{3,-\frac{13}{3}\right\}$.