Let's solve the equation \(-p \cdot (d + z) = -2z + 59\).

### Step-by-Step Solution:

1. **Expand the left-hand side**:
   \[
   -p \cdot (d + z) = -p \cdot d - p \cdot z
   \]
   So, the equation becomes:
   \[
   -p \cdot d - p \cdot z = -2z + 59
   \]

2. **Rearrange the equation**: 
   We want to isolate terms involving \(z\). First, bring all \(z\)-terms to one side.
   \[
   -p \cdot d = -2z + p \cdot z + 59
   \]
   Factor out \(z\) from the right-hand side:
   \[
   -p \cdot d = z(-2 + p) + 59
   \]

3. **Isolate \(z\)**:
   Subtract 59 from both sides:
   \[
   -p \cdot d - 59 = z(-2 + p)
   \]
   Finally, solve for \(z\):
   \[
   z = \frac{-p \cdot d - 59}{p - 2}
   \]

Thus, the solution for \(z\) in terms of \(p\) and \(d\) is:
\[
z = \frac{-p \cdot d - 59}{p - 2}
\]

Let me know if you would like more details or clarifications on any step!

### Related Questions:
1. What happens if \(p = 2\) in this equation?
2. Can you find the value of \(z\) if \(p = 3\) and \(d = 5\)?
3. How does the solution for \(z\) change if the equation is altered slightly, like \(+2z\) instead of \(-2z\)?
4. Can this equation be solved if \(p = 0\)? If so, how?
5. How would you graph the equation for different values of \(p\) and \(d\)?

### Tip:
Always check if the denominator could potentially lead to division by zero when solving rational expressions.

This guide explains how to solve the equation -p(d + z) = -2z + 59 step-by-step. It involves key algebraic techniques such as distribution and isolating variables. Perfect for high school students studying Algebra in Grades 9-11.

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