To solve the equation for \( z \):

Given equation:

\[
\frac{5}{3}(z + 1) = 3 - \frac{2}{3}z
\]

### Step 1: Distribute \( \frac{5}{3} \) on the left side

\[
\frac{5}{3} \cdot z + \frac{5}{3} \cdot 1 = \frac{5}{3}z + \frac{5}{3}
\]

So the equation becomes:

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

### Step 2: Eliminate the fractions by multiplying the entire equation by 3

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

This simplifies to:

\[
5z + 5 = 9 - 2z
\]

### Step 3: Collect all terms involving \( z \) on one side of the equation

\[
5z + 2z = 9 - 5
\]

\[
7z = 4
\]

### Step 4: Solve for \( z \)

\[
z = \frac{4}{7}
\]

So, the solution is:

\[
\boxed{z = \frac{4}{7}}
\]

Would you like any details on the solution or have any other questions?

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Here are 5 related questions:

1. How do you solve equations with variables on both sides?
2. What is the process to eliminate fractions in an equation?
3. How can you check if your solution to an equation is correct?
4. How do you solve an equation involving distribution and fractions?
5. What methods can be used to isolate the variable in an equation?

**Tip:** When solving equations with fractions, consider multiplying through by the denominator to simplify the equation early on.

Learn how to solve the equation 5/3(z + 1) = 3 - 2/3z step-by-step with detailed explanations. This problem involves equations with fractions, distribution, and variable isolation techniques. Suitable for students in Grades 7-9.

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