r4 parametrization

Question

I'd like to ask how to parametrize linear equations in $\mathbb{R^4}$?

I have two equations:

$x+2y-2z+w=-1$

$x+y+z-w=2$

I can't come up with how to eliminate two variables

Answer 1

Subtracting them, we get $y-3z+2w=-3$ ($\star$).

Now let $z=\lambda$ and $w=\mu$.

Plugging into $(\star)$, we obtain $y=-3+3\lambda-2\mu$ and $x=2-y-z+w=2-(-3+3\lambda-2\mu)-\lambda+\mu=5-4\lambda+3\mu$.

Hence a parametric equation is given by

$$\color{red}{\begin{pmatrix}x \\y\\z\\w\end{pmatrix}=\begin{pmatrix}5 \\-3\\0\\0\end{pmatrix}+\lambda\begin{pmatrix}-4 \\3\\1\\0\end{pmatrix} +\mu\begin{pmatrix}3 \\-2\\0\\1\end{pmatrix}}, \lambda,\mu\in\mathbf{R}$$