Question

1. \\(\frac{5}{x^2} = \frac{1}{x} - \frac{1}{5x^2}\\)

Explanation:

Step1: Eliminate denominators

Multiply each term by \(5x^2\) (the least common denominator of \(x^2\), \(x\), and \(5x^2\)) to get rid of the fractions:
\(5x^2 \cdot \frac{5}{x^2} = 5x^2 \cdot \frac{1}{x} - 5x^2 \cdot \frac{1}{5x^2}\)
Simplify each term:
\(25 = 5x - 1\)

Step2: Solve for \(x\)

Add 1 to both sides:
\(25 + 1 = 5x - 1 + 1\)
\(26 = 5x\)

Divide both sides by 5:
\(x = \frac{26}{5}\)

Check for extraneous solutions: Substitute \(x = \frac{26}{5}\) back into the original equation. The denominators \(x^2\), \(x\), and \(5x^2\) are non - zero (since \(x=\frac{26}{5}
eq0\)), so it is a valid solution.

Answer:

\(x = \frac{26}{5}\)