Parametrize a system of two equations and four variables.

Question

I'm working on the following problem from Ideals, Varieties, and Algorithms.

Parametrize all solutions of the linear equations

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

I started by creating the matrix \begin{bmatrix} 1 & 2 & -2 & 1 &| &-1 \\ 1 & 1 & 1 & -1 &| & 2 \end{bmatrix}

and reducing it to

\begin{bmatrix} 1 & 0 & 4 & -3 &| &5 \\ 0 & 1 & -3 & 2 &| & -3 \end{bmatrix}

Which gives me $$x+4z-3w = 5 \\y-3z+2w=-3$$

Do I have to have two parameters, $t$ and $u$, to parametrize this or can I use just one, $t$?

Answer 1

You have two columns in the reduced coefficient matrix with no pivots, so you'll need two parameters, corresponding to the two "free variables", $z$ and $w$. Rewrite those last two equations to solve for $x$ and $y$, your two "bound variables":

$$x=5-4z+3w\\ y=3+3z-2w$$

Now, if you let $z=t$, and $w=u$, then we have:

$$\left[\begin{matrix}x\\y\\z\\w \end{matrix}\right] = \left[\begin{matrix}5\\3\\0\\0 \end{matrix}\right] + \left[\begin{matrix}-4\\3\\1\\0 \end{matrix}\right]t + \left[\begin{matrix}3\\-2\\0\\1 \end{matrix}\right]u$$