The variables \( x \), \( y \), and \( z \) are not necessarily equal in the parametric equations. They represent coordinates in a three-dimensional space that depend on the parameter \( t \). Let’s clarify this:

### The Parametric Equations
The original parametric equations are:
\[
x = -1 + 2t, \quad y = 2 - 3t, \quad z = 4 + t
\]

Here, \( t \) is a parameter that varies, and for each value of \( t \), you get a corresponding point \( (x, y, z) \) in 3D space. The equations describe a line in three-dimensional space, with each coordinate \( x \), \( y \), and \( z \) changing according to \( t \).

### Why They Are Not Equal
In the parametric equations, \( x \), \( y \), and \( z \) are functions of \( t \), and these functions are different. Thus, \( x \), \( y \), and \( z \) do not necessarily equal each other.

### What Was Derived
When we eliminated the parameter \( t \), we derived two equations that relate \( x \), \( y \), and \( z \):

1. \( 3x + 2y = 1 \)
2. \( x = 2z - 9 \)

These equations describe the relationship between \( x \), \( y \), and \( z \) on the line in 3D space, but they don't imply that \( x = y = z \). Instead, they describe constraints that these variables must satisfy simultaneously.

For example:
- \( x \) is expressed in terms of \( z \) as \( x = 2z - 9 \).
- \( y \) is related to \( x \) through \( 3x + 2y = 1 \).

Thus, \( x \), \( y \), and \( z \) take specific values that satisfy both these equations, but they are not equal to each other.

### Conclusion
The variables \( x \), \( y \), and \( z \) are not equal; they are related through the equations we derived after eliminating the parameter \( t \). Each variable depends on the other two in a specific way determined by those relationships.

Would you like further clarification or have any additional questions?

### Related Questions:
1. How can you visualize the line represented by these parametric equations?
2. What is the significance of the relationship \( 3x + 2y = 1 \) in the context of geometry?
3. How do you find the point of intersection between this line and a plane in 3D space?
4. Can the parametric equations represent different geometric objects other than lines?
5. What are the conditions for \( x \), \( y \), and \( z \) to be equal in parametric equations?

### Tip:
When dealing with parametric equations, always remember that the variables are linked through the parameter, and the relationships between them define the geometric object (like a line or curve) in space.

Explore the reasons why the variables x, y, and z are not necessarily equal in parametric equations. Learn about their dependencies on the parameter t and the derived equations describing their relationships. Suitable for advanced high school students and anyone interested in 3D geometry and linear algebra.

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