Let's solve the equation:

\[
\frac{x}{3} - 2 = \frac{5}{3}x + 7
\]

### Step 1: Eliminate the fractions by multiplying the entire equation by 3.

\[
3 \left(\frac{x}{3} - 2\right) = 3 \left(\frac{5}{3}x + 7\right)
\]

This simplifies to:

\[
x - 6 = 5x + 21
\]

### Step 2: Move all the terms involving \(x\) to one side and the constants to the other side.

Subtract \(5x\) from both sides:

\[
x - 5x - 6 = 21
\]

This simplifies to:

\[
-4x - 6 = 21
\]

### Step 3: Solve for \(x\).

Add 6 to both sides:

\[
-4x = 27
\]

Now divide by \(-4\):

\[
x = \frac{27}{-4} = -\frac{27}{4}
\]

So, the solution is:

\[
x = -\frac{27}{4}
\]

Would you like a detailed explanation of any step or further clarification?

Here are some related questions you might find helpful:
1. How do you solve equations involving multiple fractions?
2. How can you check if your solution to a linear equation is correct?
3. What is the difference between moving terms and isolating variables?
4. Can you explain how to solve equations with different variables?
5. How do you handle equations with absolute values?

**Tip:** Always double-check your solution by plugging it back into the original equation to verify if both sides are equal.

Learn how to solve the linear equation x/3 - 2 = 5/3 x + 7 with a clear step-by-step breakdown. This problem focuses on key algebraic concepts like handling fractions and isolating variables. Suitable for students in Grades 8-10.

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