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 solve the equation

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

we first identify the common denominator. The common denominator for the fractions involved is \(x^2\).

Now, let's multiply every term in the equation by \(x^2\) to eliminate the fractions:

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

This gives us:

  1. The first term: \[ x^2 \cdot \frac{1}{x^2} = 1, \]

  2. The second term: \[ -x^2 \cdot x = -x^3, \]

  3. The third term: \[ x^2 \cdot \frac{1}{x} = x, \]

  4. The fourth term (on the right): \[ x^2 \cdot \frac{5}{x^2} = 5, \]

  5. The fifth term (on the right, same as the second term): \[ -x^2 \cdot x = -x^3. \]

Putting it all together, the equation becomes:

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

Now we simplify:

  1. We observe that the \(-x^3\) terms on both sides cancel out:

The equation simplifies to:

\[ 1 + x = 5. \]

To isolate \(x\), we subtract 1 from both sides:

\[ x = 5 - 1, \] \[ x = 4. \]

Thus, the equivalent expression after multiplying by the common denominator has been demonstrated, and we solved for \(x\). The final answer is \(x = 4\).