To solve the equation \( Y = \frac{3x}{z - b} \) for \( z \), follow these steps:

### Step-by-step solution:
1. Start with the given equation:
   \[
   Y = \frac{3x}{z - b}
   \]

2. Multiply both sides of the equation by \( z - b \) to eliminate the denominator:
   \[
   Y(z - b) = 3x
   \]

3. Distribute \( Y \) on the left-hand side:
   \[
   Yz - Yb = 3x
   \]

4. Add \( Yb \) to both sides to isolate the \( Yz \) term:
   \[
   Yz = 3x + Yb
   \]

5. Finally, divide both sides by \( Y \) to solve for \( z \):
   \[
   z = \frac{3x + Yb}{Y}
   \]

Thus, the solution for \( z \) is:
\[
z = \frac{3x + Yb}{Y}
\]

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Would you like more details on this solution or have any other questions?

### Related questions:
1. How would you solve for \( x \) instead of \( z \) in the same equation?
2. What happens to the equation if \( b = 0 \)?
3. Can you explain how to solve for \( b \) in terms of \( z \)?
4. How does the equation change if \( Y \) is negative?
5. What if \( z - b = 0 \)? What does that imply about the relationship between \( x \), \( z \), and \( b \)?

### Tip:
When dealing with equations involving fractions, multiplying both sides by the denominator is often the first step to eliminate the fraction and simplify the equation.

Learn how to solve for z in the rational equation Y = 3x / (z - b) with a detailed, step-by-step explanation. This problem involves key algebraic concepts, such as isolating variables and handling rational expressions. Suitable for high school students in Grades 9-12.

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