To solve the equation \(\frac{p}{x} = \frac{q}{x - 2}\), we can use cross-multiplication:

\[
p(x - 2) = qx
\]

Now, distribute the \(p\) on the left-hand side:

\[
p(x - 2) = px - 2p
\]

So, the equation becomes:

\[
px - 2p = qx
\]

Next, let's move all terms involving \(x\) to one side of the equation. Subtract \(qx\) from both sides:

\[
px - qx = 2p
\]

Factor out \(x\) on the left-hand side:

\[
x(p - q) = 2p
\]

Finally, solve for \(x\) by dividing both sides by \((p - q)\) (assuming \(p \neq q\)):

\[
x = \frac{2p}{p - q}
\]

Thus, the solution is:

\[
x = \frac{2p}{p - q}
\]

Would you like more details or further clarification?

Here are 5 related questions:
1. What happens if \(p = q\)?
2. How would the solution change if the equation had additional terms?
3. Can you solve a similar equation for specific values of \(p\) and \(q\)?
4. What if the variables were squared, i.e., \(\frac{p}{x^2} = \frac{q}{(x-2)^2}\)?
5. How would you solve this if \(p\) or \(q\) were negative?

**Tip:** Always check for restrictions in the denominator to avoid dividing by zero (e.g., \(x \neq 0\) and \(x \neq 2\) in this problem).

Learn how to solve the rational equation (p/x) = (q/(x-2)) using cross multiplication. This problem involves key algebraic concepts like rational expressions, cross-multiplication, and solving linear equations. Suitable for students in Grades 8-10.

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