<s>[INST] $3$ lines $4$ variables linear equation gaussian So I'm currently taking a Linear Algebra class and am stuck on a problem.
I have the equations:
$$\begin{cases}\begin{align}&x + 2y - z + 3t = 3\\
  &2x + 4y + 4z + 3t = 9\\
&3x + 6y - z + 8t = 10
\end{align}\end{cases}$$
I'm not sure on how to solve this with the $4$ variables and the $3$ lines. I can do a $3 \times 3$ but am struggling with this one. I'm not asking for someone to solve it, I'm asking for a point in the right direction.
 [/INST] $$\left[
  \begin{array}{cccc|c}
  1 & 2 & -1 & 3  & 3 \\
  2 & 4 &  4 & 3  & 9 \\
  3 & 6 & -1 & 8  & 10 \\
  \end{array}
  \right]$$
$$\left[
  \begin{array}{cccc|cc}
  1 & 2 & -1 & 3 &  3 \\
  0 & 0 &  6 & -3  & 3 \\
  0 & 0 & 2 & -1  & 1 \\
  \end{array}
  \right]$$
$$\left[
  \begin{array}{cccc|cc}
  1 & 2 & -1 & 3 &  3 \\
  0 & 0 &  2 & -1  & 1 \\
  0 & 0 & 0 & 0  & 0 \\
  \end{array}
  \right]$$
$$\left[
  \begin{array}{cccc|cc}
  1 & 2 & 0 & 2.5 &  3.5 \\
  0 & 0 &  2 & -1  & 1 \\
  0 & 0 & 0 & 0  & 0 \\
  \end{array}
  \right]$$
$$x+2y+2.5t=3.5\\2z-t=1$$you have infinitely many solutions
</s>