https://file.aiquickdraw.com/mathbot/file/172710138543244wysffw.jpg

The equation presented in the image is:

\[
-p \cdot (d + z) = -2z + 59
\]

We are asked to solve for \( z \). Let's solve the equation step by step.

### Step 1: Expand both sides of the equation
Distribute the \( -p \) on the left-hand side:

\[
-p \cdot d - p \cdot z = -2z + 59
\]

This simplifies to:

\[
-pd - pz = -2z + 59
\]

### Step 2: Collect all \( z \)-terms on one side
Move the \( pz \) and \( -2z \) terms together:

\[
-pd = -2z + pz + 59
\]

### Step 3: Factor out \( z \)
Now, isolate \( z \) by factoring:

\[
-pd - 59 = z(-2 + p)
\]

### Step 4: Solve for \( z \)
Finally, divide both sides by \( -2 + p \) to solve for \( z \):

\[
z = \frac{-pd - 59}{-2 + p}
\]

Thus, the solution is:

\[
z = \frac{-pd - 59}{p - 2}
\]

Let me know if you'd like further details or clarifications!

---

Here are 5 related questions to expand on the solution:

1. How would the solution change if \( p = 2 \)?
2. What happens if \( d = 0 \) in the solution?
3. How would you solve if the right-hand side was \( -2z - 59 \) instead?
4. How can this solution be interpreted in a real-world application?
5. How does the presence of a constant like \( 59 \) affect solving linear equations?

**Tip**: Always check if the denominator in a solution can lead to division by zero, as it may restrict possible solutions.

Solve for z in the equation -p(d + z) = -2z + 59. Assume the equation has a solution for z.

Learn how to solve the linear equation -p(d + z) = -2z + 59. This problem covers algebraic concepts such as distributing and collecting like terms. Suitable for students in Grades 8-10.

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