To identify the equivalent expression in the equation
\[ \frac{1}{x^2} - x + \frac{1}{x} = \frac{5}{x^2} - x \]
we can start by eliminating the fractions by multiplying through by the common denominator, which in this case is \(x^2\).
Let's do that step-by-step.
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Identify the common denominator: The common denominator for all terms in the equation is \(x^2\).
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Multiply the entire equation by \(x^2\):
\[ x^2 \left(\frac{1}{x^2}\right) - x^2(x) + x^2\left(\frac{1}{x}\right) = x^2\left(\frac{5}{x^2}\right) - x^2(x) \]
- Distributing \(x^2\) across each term:
- \(\frac{1}{x^2} \cdot x^2 = 1\)
- \(-x^2 \cdot x = -x^3\)
- \(\frac{1}{x} \cdot x^2 = x\)
- \(\frac{5}{x^2} \cdot x^2 = 5\)
- \(-x^2 \cdot x = -x^3\)
Putting it all together, we have:
\[ 1 - x^3 + x = 5 - x^3 \]
- Now we can simplify the equation:
Cancelling out \(-x^3\) from both sides, we get:
\[ 1 + x = 5 \]
- Solving for \(x\):
Subtract \(1\) from both sides:
\[ x = 5 - 1 = 4 \]
Thus, the equivalent expression in the equation when we multiply by the common denominator is:
\[ 1 - x^3 + x = 5 - x^3 \]
And the equation simplifies neatly to \(x = 4\).