The equation you've provided is:

\[
\frac{2}{x} = \frac{10}{x - 6}
\]

To solve this equation, we can cross-multiply:

\[
2(x - 6) = 10x
\]

Now, let's distribute the 2 on the left-hand side:

\[
2x - 12 = 10x
\]

Next, we'll move all terms involving \(x\) to one side and constants to the other:

\[
2x - 10x = 12
\]

Simplifying:

\[
-8x = 12
\]

Now, solve for \(x\):

\[
x = \frac{12}{-8} = -\frac{3}{2}
\]

So, the solution is:

\[
x = -\frac{3}{2}
\]

Do you want more details on the solution process or have any questions?

Here are five related questions to expand on this concept:
1. How would the solution change if the equation were \(\frac{2}{x} = \frac{10}{x + 6}\)?
2. What happens if both sides of the equation have different variables, such as \(\frac{2}{x} = \frac{10}{y - 6}\)?
3. Can you solve \(\frac{3}{x + 1} = \frac{9}{x - 5}\)?
4. How do you check if your solution is correct by substituting back into the original equation?
5. What if you had an extra constant, such as \(\frac{2}{x} + 5 = \frac{10}{x - 6}\)?

**Tip:** When cross-multiplying fractions, ensure the variables in the denominator do not make the denominator equal to zero to avoid undefined solutions.

Learn how to solve the rational equation 2/x = 10/(x - 6) using cross-multiplication and simplifying linear equations. Follow the step-by-step guide suitable for students in Grades 8-10.

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