Let's solve the equation step by step:

The equation is:

\[
16z + 29z = pz - v
\]

### Step 1: Combine like terms on the left-hand side

\[
(16z + 29z) = 45z
\]

So, the equation becomes:

\[
45z = pz - v
\]

### Step 2: Move all terms involving \( z \) to one side of the equation

Subtract \( pz \) from both sides:

\[
45z - pz = -v
\]

### Step 3: Factor out \( z \) on the left-hand side

\[
z(45 - p) = -v
\]

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

To solve for \( z \), divide both sides by \( (45 - p) \), assuming \( p \neq 45 \):

\[
z = \frac{-v}{45 - p}
\]

### Final Answer:

\[
z = \frac{-v}{45 - p}
\]

Let me know if you'd like further details on any step.

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Here are five related questions:
1. How would the solution change if \( p = 45 \)?
2. Can you solve a similar equation with different coefficients, like \( 10z + 5z = pz - v \)?
3. How would you check the solution once you have \( z \)?
4. What happens if \( v = 0 \)?
5. How can you solve the equation if it includes fractions?

**Tip:** Always combine like terms and isolate variables to one side for easier solving.

Learn how to solve the linear equation 16z + 29z = pz - v using algebraic techniques. This detailed step-by-step guide walks through combining like terms, isolating variables, and factoring, making it suitable for students in Grades 9-10.

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