To understand how a term like \(-3z\) might appear in an equation, let's consider a few possible scenarios:

### 1. **Subtraction or Negative Multiplication**:
   - Suppose you start with a term like \(5z\) and subtract \(8z\) from it: 
     \[
     5z - 8z = -3z
     \]
   - Here, subtracting \(8z\) from \(5z\) results in \(-3z\).

### 2. **Multiplying by a Negative Coefficient**:
   - If you multiply a positive variable \(z\) by \(-3\), you get:
     \[
     -3 \times z = -3z
     \]

### 3. **Simplifying an Expression**:
   - Sometimes, in the process of simplifying expressions or equations, you might combine like terms and end up with a \(-3z\).
   - For example:
     \[
     (2z - 5z) + z = -3z + z = -2z
     \]
   - Initially, you get \(-3z\) when combining \(2z\) and \(-5z\), though further simplification can occur.

### 4. **Solving an Equation**:
   - When solving an equation, isolating the variable might lead to \(-3z\).
     \[
     6 - 9z = 3 \quad \Rightarrow \quad -9z = 3 - 6 = -3 \quad \Rightarrow \quad z = \frac{-3}{-9} = \frac{1}{3}
     \]
   - Here, moving terms around can result in a \(-3z\) term.

### 5. **Result of an Operation**:
   - Sometimes, an operation within an equation or expression directly results in a \(-3z\), particularly when combining or rearranging terms during calculations.

Would you like further clarification on any of these scenarios?

### Follow-up Questions:
1. Can you show a specific example where \(-3z\) appears?
2. Are you dealing with an equation where you need help simplifying?
3. Do you need to understand how to handle negative coefficients in algebra?
4. Would you like to practice solving equations that involve negative terms?
5. Are there other algebraic concepts you're struggling with?

**Tip**: When working with negative signs, carefully track their effect on the terms in your equations to avoid mistakes!

Explore how the term -3z appears in algebraic equations. Learn key concepts such as subtraction, negative multiplication, and expression simplification. Suitable for students in Grades 7-9 looking to strengthen their understanding of algebraic manipulation.

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