The question involves three planes in 3D space that have no solution, meaning the planes are either parallel or do not intersect at a common point.

Given system of equations:

• $x - y - z = 0.5$
• $2x - y - z = 1.5$

You are asked to determine the third equation such that all three planes in the system have no solutions, which geometrically means they might either be parallel or intersect in a manner that results in no common intersection point.

Steps to Solve:

1. Write down the equations:

   • $x - y - z = 0.5$ (Equation 1)
   • $2x - y - z = 1.5$ (Equation 2)

2. Analyze parallelism:

   • We look for a third equation where the three planes are parallel (or nearly parallel) or inconsistent.

3. Subtract Equation 1 from Equation 2 to simplify: $(2x - y - z) - (x - y - z) = 1.5 - 0.5$ Simplifying: $x = 1$ This shows that the planes are close to parallel but shifted slightly, as the difference between the two equations yields a constant value.

4. Geometrically incompatible plane: To ensure the system has no solution, the third equation must not allow an intersection. Hence, a third plane must also not be parallel but must also not intersect the previous planes in a consistent manner. This suggests that option (C) might be a suitable candidate.

Answer:

The correct choice is (C) $2x + 2y - 2z = 1$, as it would create an inconsistency among the planes, resulting in no common point of intersection.

Would you like more detailed steps on this, or any further questions?

Here are five questions for deeper exploration:

1. How do we generally determine if planes in 3D space are parallel?
2. Why does subtracting the equations help in solving systems of planes?
3. What are the possible geometric configurations for systems of three planes?
4. How can we tell from the equations whether the planes intersect or are parallel?
5. How could we use matrix methods like Gaussian elimination to solve such systems?

Tip: Always check for geometric consistency when solving systems involving planes—parallel planes and inconsistent systems often indicate no solutions.