Question

1. solve for m:

\frac{1}{m^2 - 3m} - \frac{1}{m} = -1

Explanation:

Step1: Factor the denominator

First, factor \( m^2 - 3m \) as \( m(m - 3) \). So the equation becomes \(\frac{1}{m(m - 3)} - \frac{1}{m} = -1\).

Step2: Find a common denominator

The common denominator for the left - hand side is \( m(m - 3) \). Rewrite \(\frac{1}{m}\) with the common denominator: \(\frac{1}{m}=\frac{m - 3}{m(m - 3)}\). Now the equation is \(\frac{1}{m(m - 3)}-\frac{m - 3}{m(m - 3)}=-1\).

Step3: Combine the fractions

Subtract the fractions on the left - hand side: \(\frac{1-(m - 3)}{m(m - 3)}=-1\). Simplify the numerator: \(1-(m - 3)=1 - m + 3=4 - m\). So we have \(\frac{4 - m}{m(m - 3)}=-1\).

Step4: Cross - multiply

Multiply both sides by \( m(m - 3) \) to get \( 4 - m=-m(m - 3) \).

Step5: Expand the right - hand side

Expand \( -m(m - 3) \) to get \( -m^2+3m \). So the equation is \( 4 - m=-m^2 + 3m \).

Step6: Rearrange into standard quadratic form

Bring all terms to one side: \( m^2-4m + 4 = 0 \).

Step7: Factor the quadratic equation

The quadratic \( m^2-4m + 4 \) factors as \( (m - 2)^2=0 \).

Step8: Solve for m

Take the square root of both sides: \( m - 2 = 0 \), so \( m = 2 \). We need to check for extraneous solutions. When \( m = 2 \), the original denominators \( m^2-3m=4 - 6=-2
eq0\) and \( m = 2
eq0\), so \( m = 2\) is a valid solution.

Answer:

\( m = 2 \)