Identify the equivalent expression in the equation 1/x^2-x+1/x=5/x^2-x and demonstrate multiplying by the common denominator

1 answer

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.

  1. Identify the common denominator: The common denominator for all terms in the equation is \(x^2\).

  2. Multiply the entire equation by \(x^2\):

\[ x^2 \left(\frac{1}{x^2}\right) - x^2(x) + x^2\left(\frac{1}{x}\right) = x^2\left(\frac{5}{x^2}\right) - x^2(x) \]

  1. 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:

\[ 1 - x^3 + x = 5 - x^3 \]

  1. Now we can simplify the equation:

Cancelling out \(-x^3\) from both sides, we get:

\[ 1 + x = 5 \]

  1. 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 \(x = 4\).