https://file.aiquickdraw.com/mathbot/file/1726835223287g7o76vnq.jpg

The equation shown in the image is:

\[
2(4z - 1) = 3(z + 2)
\]

Let's solve this step by step.

### Step 1: Distribute the constants on both sides.
Expand the equation by distributing \(2\) and \(3\) to the terms inside the parentheses:

\[
2 \cdot (4z - 1) = 3 \cdot (z + 2)
\]

This becomes:

\[
8z - 2 = 3z + 6
\]

### Step 2: Move the variables to one side.
To get all the \(z\)-terms on one side, subtract \(3z\) from both sides:

\[
8z - 3z - 2 = 6
\]

This simplifies to:

\[
5z - 2 = 6
\]

### Step 3: Isolate \(z\).
Now, add \(2\) to both sides to move the constant to the other side:

\[
5z = 6 + 2
\]

\[
5z = 8
\]

### Step 4: Solve for \(z\).
Finally, divide both sides by 5 to solve for \(z\):

\[
z = \frac{8}{5}
\]

So, the solution is:

\[
z = \frac{8}{5}
\]

Would you like further details on any of these steps?

---

Here are 5 related questions to deepen understanding:
1. How do you check if \( z = \frac{8}{5} \) is the correct solution?
2. What happens if you move the constants to the other side first?
3. How would the solution process change if the equation was \(2(4z - 1) = 3(z + 3)\)?
4. Can you solve similar equations where the distribution involves negative numbers?
5. How do you interpret equations graphically, and what does the solution \( z = \frac{8}{5} \) represent on a graph?

**Tip:** Always perform inverse operations carefully to avoid sign mistakes during algebraic manipulations.

Learn how to solve the linear equation 2(4z - 1) = 3(z + 2) with a detailed step-by-step solution. This problem covers key algebraic concepts, including the distributive property and isolating variables. Suitable for students in Grades 7-9.

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