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Let's work through this equation step-by-step.

The given equation is:
\[
\frac{5}{3}x + \frac{1}{3}x = 13 \frac{1}{3} + \frac{8}{3}x
\]

### Step 1: Combine like terms on each side
On the left side:
\[
\frac{5}{3}x + \frac{1}{3}x = \frac{6}{3}x = 2x
\]

So the equation simplifies to:
\[
2x = 13 \frac{1}{3} + \frac{8}{3}x
\]

### Step 2: Convert \(13 \frac{1}{3}\) to an improper fraction
\[
13 \frac{1}{3} = \frac{40}{3}
\]

Now the equation becomes:
\[
2x = \frac{40}{3} + \frac{8}{3}x
\]

### Step 3: Move the \( \frac{8}{3}x \) term to the left side by subtracting it from both sides
\[
2x - \frac{8}{3}x = \frac{40}{3}
\]

To combine \(2x\) and \(\frac{8}{3}x\), rewrite \(2x\) as \(\frac{6}{3}x\):
\[
\frac{6}{3}x - \frac{8}{3}x = \frac{-2}{3}x
\]

So the equation simplifies to:
\[
\frac{-2}{3}x = \frac{40}{3}
\]

### Step 4: Solve for \(x\) by dividing both sides by \(-\frac{2}{3}\)
Dividing by \(-\frac{2}{3}\) is equivalent to multiplying by \(-\frac{3}{2}\):
\[
x = \frac{40}{3} \cdot -\frac{3}{2} = -20
\]

### Final Answer
\[
x = -20
\]

Would you like further clarification on any of these steps, or do you need help with similar problems? Here are five additional questions that build on this problem:

1. How do you convert a mixed number to an improper fraction?
2. Why is it helpful to combine like terms before moving terms from one side of the equation to the other?
3. How do you simplify expressions with fractional coefficients?
4. What are alternative ways to solve for a variable when fractions are involved?
5. How can we check if our solution for \(x\) is correct?

**Tip:** Always double-check your work by substituting your answer back into the original equation.

Solve the equation \( \frac{5}{3}x + \frac{1}{3}x = 13 \frac{1}{3} + \frac{8}{3}x \) step-by-step.

This guide walks you through solving the linear equation \( \frac{5}{3}x + \frac{1}{3}x = 13 \frac{1}{3} + \frac{8}{3}x \), including steps on combining like terms and handling mixed numbers. Suitable for students in Grades 8-10.

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