The equation is:

\[
-4z + 1 = bz + c
\]

To solve for \(z\), follow these steps:

1. **Move all terms involving \(z\) to one side**:  
   Subtract \(bz\) from both sides:
   \[
   -4z - bz + 1 = c
   \]
   Combine the \(z\)-terms:
   \[
   (-4 - b)z + 1 = c
   \]

2. **Isolate the term with \(z\)**:  
   Subtract 1 from both sides:
   \[
   (-4 - b)z = c - 1
   \]

3. **Solve for \(z\)**:  
   Divide both sides by \((-4 - b)\):
   \[
   z = \frac{c - 1}{-4 - b}
   \]

Thus, the solution for \(z\) is:

\[
z = \frac{c - 1}{-4 - b}
\]

Let me know if you need more details or clarification.

Here are 5 related questions:
1. What happens if \(b = -4\) in the equation?
2. Can you solve for \(b\) or \(c\) if \(z\) is known?
3. How would you interpret this equation graphically?
4. What if \(c = 1\)? What does this imply for the solution?
5. How does the solution change if the equation is \(4z + 1 = bz + c\)?

**Tip:** Always check if any value for variables might cause division by zero when solving equations!

Learn how to solve the linear equation -4z + 1 = bz + c using step-by-step algebraic methods. This problem involves isolating the variable z and manipulating both sides of the equation. Suitable for students in Grades 8-10.

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