eliminate parameter t from equation L of x=-1+2t, y=2-3t, z=4+t

To eliminate the parameter \( t \) from the given parametric equations:

\[
x = -1 + 2t, \quad y = 2 - 3t, \quad z = 4 + t
\]

we can solve each equation for \( t \) and then express them in terms of \( x \), \( y \), and \( z \).

### Step 1: Solve each equation for \( t \):

1. From \( x = -1 + 2t \):
   \[
   x + 1 = 2t \quad \Rightarrow \quad t = \frac{x + 1}{2}
   \]

2. From \( y = 2 - 3t \):
   \[
   y - 2 = -3t \quad \Rightarrow \quad t = \frac{2 - y}{3}
   \]

3. From \( z = 4 + t \):
   \[
   t = z - 4
   \]

### Step 2: Set these expressions for \( t \) equal to each other:

\[
\frac{x + 1}{2} = \frac{2 - y}{3} = z - 4
\]

### Step 3: Eliminate \( t \) to find the relationship between \( x \), \( y \), and \( z \):

1. Equating the first two expressions:
   \[
   \frac{x + 1}{2} = \frac{2 - y}{3}
   \]
   Cross-multiply to get:
   \[
   3(x + 1) = 2(2 - y)
   \]
   Simplify:
   \[
   3x + 3 = 4 - 2y \quad \Rightarrow \quad 3x + 2y = 1
   \]

2. Equating the first and third expressions:
   \[
   \frac{x + 1}{2} = z - 4
   \]
   Cross-multiply to get:
   \[
   x + 1 = 2(z - 4) \quad \Rightarrow \quad x = 2z - 9
   \]

So, the final system of equations without the parameter \( t \) is:

\[
\begin{cases}
3x + 2y = 1 \\
x = 2z - 9
\end{cases}
\]

This represents the relationship between \( x \), \( y \), and \( z \) after eliminating the parameter \( t \).

Would you like further details or have any questions?

### Related Questions:
1. How would you solve for \( y \) in terms of \( x \) and \( z \) using the final system?
2. What is the geometric interpretation of the system of equations in 3D space?
3. How do you parametrize a line given two points in space?
4. How can you find the direction vector of the line from the parametric equations?
5. What is the significance of eliminating the parameter \( t \) in parametric equations?

### Tip:
Always cross-check the solutions by substituting back into the original equations to ensure consistency.

Learn how to eliminate the parameter t from the given parametric equations x = -1 + 2t, y = 2 - 3t, z = 4 + t. Understand the process step-by-step with detailed solutions and equations. Suitable for advanced high school students and those studying parametric equations.

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