Question

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

Explanation:

Step 1: Eliminate the fraction denominators

To solve the equation \(\frac{1}{x^{2}}-\frac{1}{x}=\frac{5}{x^{2}}\), we first note that \(x
eq0\) (since division by zero is undefined). Multiply each term by \(x^{2}\) to clear the denominators:

$$ x^{2}\cdot\frac{1}{x^{2}}-x^{2}\cdot\frac{1}{x}=x^{2}\cdot\frac{5}{x^{2}} $$

Simplifying each term, we get:

$$ 1 - x=5 $$

Step 2: Solve for \(x\)

Now, solve the linear equation \(1 - x = 5\). Subtract 1 from both sides:

$$ - x=5 - 1 $$

$$ - x = 4 $$

Multiply both sides by - 1:

$$ x=-4 $$

Step 3: Check the solution

Substitute \(x = - 4\) back into the original equation:
Left - hand side: \(\frac{1}{(-4)^{2}}-\frac{1}{-4}=\frac{1}{16}+\frac{1}{4}=\frac{1 + 4}{16}=\frac{5}{16}\)
Right - hand side: \(\frac{5}{(-4)^{2}}=\frac{5}{16}\)
Since the left - hand side equals the right - hand side when \(x=-4\), the solution is valid.

Answer:

\(x = - 4\)