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:

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.