Question

Solve the equation.
StartFraction 36 Over 3 x minus 15 EndFraction plus StartFraction 1 Over x minus 5 EndFraction equals StartFraction 13 Over 3 EndFraction

Answer 1

To solve the equation

\[
\frac{36}{3x - 15} + \frac{1}{x - 5} = \frac{13}{3},
\]

let's first simplify it step by step.

1. **Simplify \(3x - 15\):**
\[
3x - 15 = 3(x - 5).
\]
Therefore, we can rewrite the first fraction:
\[
\frac{36}{3x - 15} = \frac{36}{3(x - 5)} = \frac{12}{x - 5}.
\]

2. **Rewrite the equation with the simplified fraction:**
\[
\frac{12}{x - 5} + \frac{1}{x - 5} = \frac{13}{3}.
\]
Combine the fractions on the left:
\[
\frac{12 + 1}{x - 5} = \frac{13}{x - 5}.
\]
Thus, the equation becomes:
\[
\frac{13}{x - 5} = \frac{13}{3}.
\]

3. **Cross-multiply:**
\[
13 \cdot 3 = 13 \cdot (x - 5).
\]
Canceling out \(13\) (assuming \(x - 5 \neq 0\)):
\[
3 = x - 5.
\]

4. **Solve for \(x\):**
\[
x = 3 + 5 = 8.
\]

5. **Check for extraneous solutions:**
We need to verify that \(x = 8\) does not make any denominators zero. The original denominators were \(3x - 15\) and \(x - 5\). For \(x = 8\):
\[
3(8) - 15 = 24 - 15 = 9 \quad (\text{not zero}),
\]
\[
8 - 5 = 3 \quad (\text{not zero}).
\]

Thus, the solution is valid.

The final answer is:
\[
\boxed{8}.
\]