Question

solve for n.\\(\frac{n}{5n} = \frac{1}{n - 5}\\)

Explanation:

Step1: Simplify left - hand side

The left - hand side of the equation is $\frac{n}{5n}$. We can cancel out the common factor $n$ (assuming $n
eq0$), so $\frac{n}{5n}=\frac{1}{5}$. The equation becomes $\frac{1}{5}=\frac{1}{n - 5}$.

Step2: Cross - multiply

Cross - multiplying gives us $1\times(n - 5)=1\times5$.

Step3: Solve for n

Simplify the equation $n-5 = 5$. Add 5 to both sides of the equation: $n=5 + 5=10$.
We need to check for extraneous solutions. When $n = 0$, the original left - hand side $\frac{n}{5n}$ is undefined. When $n=10$, the left - hand side is $\frac{10}{5\times10}=\frac{10}{50}=\frac{1}{5}$, and the right - hand side is $\frac{1}{10 - 5}=\frac{1}{5}$. When $n = 5$, the right - hand side is undefined. So $n = 10$ is a valid solution.

Answer:

$n = 10$