Question

solve the rational equation: $\frac{1}{n^{2}-5n}=-\frac{1}{n - 5}$
$n = 1$
$n = 5$
$n=-1$
$n = 5,n=-1$

Explanation:

Step1: Factor the denominator

Factor $n^{2}-5n$ to $n(n - 5)$. The equation becomes $\frac{1}{n(n - 5)}=-\frac{1}{n - 5}$.

Step2: Multiply both sides by $n(n - 5)$

$1=-n$.

Step3: Solve for $n$

$n=-1$.
We need to check for extraneous solutions. When $n = 5$, the original denominators $n^{2}-5n=0$ and $n - 5=0$, so $n = 5$ is an extraneous solution. When $n=-1$, the original equation is well - defined.

Answer:

$n=-1$