To solve the equation

\[
\frac{1}{3}(x + 6) = \frac{2}{3}(x - 9),
\]

we'll start by eliminating the fractions by multiplying both sides by 3:

\[
3 \cdot \frac{1}{3}(x + 6) = 3 \cdot \frac{2}{3}(x - 9).
\]

Simplifying, we get:

\[
x + 6 = 2(x - 9).
\]

Next, distribute the 2 on the right side:

\[
x + 6 = 2x - 18.
\]

Now, subtract \(x\) from both sides to isolate the \(x\) terms:

\[
6 = x - 18.
\]

Add 18 to both sides to solve for \(x\):

\[
6 + 18 = x,
\]

\[
x = 24.
\]

Thus, the solution to the equation is:

\[
\boxed{x = 24}.
\]

Would you like more details on any step or have any questions?

Here are 5 related questions to expand your understanding:

1. What happens if you have different denominators in a similar equation?
2. How do you solve equations with variables on both sides?
3. What are some strategies for solving equations that involve distributing and combining like terms?
4. How can you check if your solution to an equation is correct?
5. What if the equation had more fractions with different denominators?

**Tip:** When solving equations with fractions, a good first step is often to eliminate the fractions by multiplying through by the least common denominator.

Learn how to solve the equation 1/3(x+6)=2/3(x-9) step-by-step with detailed explanations. This problem involves equations and fractions, suitable for students in Grades 7-9.

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