Question

Using Gaussian elimination, find all solutions to each of the given systems of equations expressing solutions as a linear combination of all vectors.

5x+2y+4z=24
2x-y+z=9
x-2y=z=8

Answer 1

I worked out all of the Gaussian elimination and ended up with:
x-2y+z=8
3y-z=-7
-z+16

I just don't know what to do from there

Answer 2

\[5x+2y+4z=24\\
2x-y+z=9\\
x-2y\color{brown}\pm z=8\]

\[\left[ \begin{array}{cc}5&2&4\\
2&-1&1\\1&-2&\color{brown}\pm1\end{array}\right]\left[\begin{array}{c}x\\y\\z
\end{array}\right] =\left[\begin{array}{c}24\\9\\8
\end{array}\right]\]
\[\left[\begin{array}{c}5&2&4\\
2&-1&1\\1&-2&\color{brown}\pm1
\end{array}\left|\begin{array}{c}24\\9\\8
\end{array}\right.\right]\]

Answer 3

I already wrote the augmented matrix, as well as used Gaussian Elimination, I do not know how to write the solutions as a linear combination

Answer 4

dont stop eliminating until you get reduced row echelon form \[\left[\begin{array}{c}1&0&0\\
0&1&0\\0&0&1
\end{array}\left|\begin{array}{c}x\\y\\z
\end{array}\right.\right]\]

Answer 5

That wasn't how I was taught. I am trying to make that equation that I have into a form like
x -3 -2
y = -3 +t 5
z 0 1

as an example, I don't know how to draw matrices on here. I need to, through the gaussian elimation I used, turn my solutions into that form