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The question asks which value of \( a \) in the equation \( z - 1 = a(1 - z) \) would result in the equation having infinitely many solutions.

### Steps:
1. Start with the given equation:
   \[
   z - 1 = a(1 - z)
   \]

2. Distribute \( a \) on the right-hand side:
   \[
   z - 1 = a - az
   \]

3. Move all terms involving \( z \) to one side and constants to the other:
   \[
   z + az = a + 1
   \]

4. Factor out \( z \) from the left-hand side:
   \[
   z(1 + a) = a + 1
   \]

5. For the equation to have infinitely many solutions, the equation must be true for all values of \( z \). This implies that the coefficient of \( z \) on the left-hand side must be equal to the constant term on the right-hand side.

   \[
   1 + a = 0
   \]

6. Solving for \( a \):
   \[
   a = -1
   \]

Thus, the value of \( a \) that makes the equation have infinitely many solutions is \( a = -1 \).

The correct answer is **B. \( a = -1 \)**.

Would you like more details, or do you have any questions?

Here are some related questions:
1. What does it mean for an equation to have infinitely many solutions?
2. How would the solution change if we were looking for no solutions?
3. How do you solve equations that involve distribution, like in this problem?
4. What is the significance of factoring in algebraic equations?
5. Can equations like this be solved graphically, and what would that look like?

**Tip:** Always check the condition that makes an equation have infinitely many solutions by ensuring both sides are identical after simplification.

Which of the following values for a in the equation z - 1 = a(1 - z) would result in the equation having infinitely many solutions?

Learn how to find the value of a in the equation z - 1 = a(1 - z) that results in infinitely many solutions. This step-by-step algebraic problem covers key concepts in algebra and equation solving, suitable for students in Grades 9-11.

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