Question

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

Answer 1

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 \(x^2\)**:

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

3. **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:

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

4. **Now we can simplify the equation**:

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

\[
1 + x = 5
\]

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 \(x = 4\).