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The question presented is related to hydraulics, which operates on the principle of Pascal's Law. This law states that pressure exerted at any point in a confined fluid is transmitted undiminished throughout the fluid in all directions.

### Breakdown:
- **Piston X** is being pushed down, exerting an additional pressure \( P_X \) on the hydraulic fluid.
- The system also includes two other pistons, **Y** and **Z**, which experience pressure changes \( P_Y \) and \( P_Z \), respectively.

### Key Principles:
- In a hydraulic system, the pressure at all points in the same fluid must be the same (assuming it is incompressible and there are no other factors like height differences affecting the pressure).
- Since \( P_X \), \( P_Y \), and \( P_Z \) are in the same fluid and connected, the pressure exerted by one piston is transmitted equally to the others unless the areas of the pistons differ.

If all the pistons have the same surface area, the pressure exerted on each piston will be the same. Therefore, the correct relationship between \( P_X \), \( P_Y \), and \( P_Z \) is:

\[
P_X = P_Y = P_Z
\]

This matches **Option B**.

### Answer: **B) \( P_X = P_Y = P_Z \)**

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Would you like a more detailed explanation of how Pascal's law applies here? Here are some related questions to consider:

1. What is Pascal's Law and how does it relate to pressure transmission in a fluid?
2. How would the relationship between pressures change if the areas of pistons Y and Z were different?
3. Can pressure in a hydraulic system be affected by the height difference between pistons?
4. What happens to the pressure distribution if the system contains a compressible fluid instead of an incompressible one?
5. How does the area of the pistons affect the force each one experiences in a hydraulic system?

**Tip:** Remember that in a hydraulic system, while pressure is constant across all points in the same fluid, the force applied to each piston depends on the piston area.

A force acts on piston X to push it down into a hydraulic cylinder. This exerts an additional pressure Px on the hydraulic fluid in the cylinder. As a result, the two other pistons, Y and Z, in the cylinder experience a change in pressure PY and PZ respectively. What is the relationship between Px, PY, and PZ?

Learn how Pascal's Law applies to a hydraulic system with three pistons and how the pressure exerted on piston X is transmitted to pistons Y and Z. This problem explores the principles of pressure transmission in a fluid, with a focus on equal pressure distribution. Suitable for students in Grades 9-12.

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