Solve the equation $\frac{2}{p-1} - \frac{1}{p-4} = 0$ for $p$.

1. Problem Description

Solve the equation $\frac{2}{p-1} - \frac{1}{p-4} = 0$ for $p$.

2. Solution Steps

First, move the second term to the right side of the equation:

$$\frac{2}{p-1} = \frac{1}{p-4}$$

Next, cross-multiply to eliminate the fractions:

$$2(p-4) = 1(p-1)$$

Expand both sides:

$$2p - 8 = p - 1$$

Subtract $p$ from both sides:

$$2p - p - 8 = p - p - 1$$

$$p - 8 = -1$$

Add 8 to both sides:

$$p - 8 + 8 = -1 + 8$$

$$p = 7$$

We should check if $p=7$ is a valid solution.

If $p = 7$, then $p-1 = 6$ and $p-4 = 3$.

Then $\frac{2}{p-1} - \frac{1}{p-4} = \frac{2}{6} - \frac{1}{3} = \frac{1}{3} - \frac{1}{3} = 0$.

So $p=7$ is a valid solution.

3. Final Answer

$p = 7$