help do this 2/3 is a fraction - 2/3z+32=z+6

Given the equation:

\[
\frac{2}{3} - \frac{2}{3}z + 32 = z + 6
\]

Let's solve this step by step.

### Step 1: Simplify the equation
The equation already has a fraction. Let's start by isolating the terms involving \( z \) on one side. First, subtract \(\frac{2}{3}\) from both sides:

\[
-\frac{2}{3}z + 32 = z + 6 - \frac{2}{3}
\]

### Step 2: Simplify the right-hand side
Combine the constants on the right-hand side:

\[
z + 6 - \frac{2}{3} = z + \left(6 - \frac{2}{3}\right) = z + \frac{18}{3} - \frac{2}{3} = z + \frac{16}{3}
\]

Now, the equation becomes:

\[
-\frac{2}{3}z + 32 = z + \frac{16}{3}
\]

### Step 3: Clear the fraction
Multiply every term in the equation by 3 to eliminate the fraction:

\[
3 \left(-\frac{2}{3}z\right) + 3 \times 32 = 3z + 16
\]

Simplifying each term, we get:

\[
-2z + 96 = 3z + 16
\]

### Step 4: Collect like terms
Add \( 2z \) to both sides to isolate \( z \) on one side of the equation:

\[
96 = 5z + 16
\]

Next, subtract 16 from both sides to isolate the term involving \( z \):

\[
80 = 5z
\]

### Step 5: Solve for \( z \)
Divide both sides by 5:

\[
z = \frac{80}{5} = 16
\]

So, the solution to the equation is:

\[
z = 16
\]

Would you like further details, or do you have any questions?

### Related Questions:
1. How do you solve linear equations that include fractions?
2. What is the general process for clearing fractions in an equation?
3. Can this method be used for equations involving more complex fractions?
4. What are some examples of equations with multiple fractions?
5. How can you check if your solution to an equation is correct?

**Tip:** When working with fractions in equations, it's often helpful to multiply through by the least common denominator (LCD) to clear the fractions and simplify the equation.

Learn how to solve the linear equation involving fractions: 2/3 - 2/3z + 32 = z + 6. This detailed step-by-step guide covers simplifying equations, clearing fractions, and solving for variables. Ideal for students in Grades 7-9.

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