Question

Solve the rational equation:
1/5 + 2/x = 1/3

​

Answer 1

To solve the rational equation

\[
\frac{1}{5} + \frac{2}{x} = \frac{1}{3},
\]

we will first eliminate the fractions by finding a common denominator. The common denominator for 5, \(x\), and 3 is \(15x\). We will multiply every term in the equation by \(15x\):

\[
15x \cdot \frac{1}{5} + 15x \cdot \frac{2}{x} = 15x \cdot \frac{1}{3}.
\]

This simplifies to:

\[
3x + 30 = 5x.
\]

Now, we can rearrange the equation to isolate \(x\). First, subtract \(3x\) from both sides:

\[
30 = 5x - 3x,
\]

which simplifies to:

\[
30 = 2x.
\]

Now, divide both sides by 2:

\[
x = 15.
\]

Thus, the solution to the equation is

\[
\boxed{15}.
\]

To verify, substitute \(x = 15\) back into the original equation:

\[
\frac{1}{5} + \frac{2}{15} = \frac{1}{3}.
\]

Finding a common denominator for the left side, which is 15, we rewrite:

\[
\frac{3}{15} + \frac{2}{15} = \frac{5}{15} = \frac{1}{3}.
\]

Both sides match, confirming that \(x = 15\) is indeed the correct solution.