I'm going through my textbook solving the practice problems, I haven't had any trouble solving systems that are already in row-echelon form, or reduced row-echelon form. However, I'm struggling with using the Gaussian and Gauss-Jordan methods to get them to this point. One of the questions I have is:
$$x_1+x_2+2x_3=8$$
$$-x_1-2x_2+3x_3=1$$
$$3x_1-7x_2+4x_3=10$$
Which as an augmented matrix would be:
$$\begin{align}
\begin{bmatrix}
1 & 2 & 3 & 8\\
-1 & -2 & 3 & 1\\
3 & -7 & 4 & 10
\end{bmatrix}
\end{align}
$$
I understand the first step, which is to take a multiple of 1, that when added would set -1 to 0, which in this case would be 1. And this would give me:
$$\begin{align}
\begin{bmatrix}
1 & 2 & 3 & 8\\
0 & -1 & 4 & 2\\
3 & -7 & 4 & 10
\end{bmatrix}
\end{align}
$$
But after this I'm lost as to what I should do next, or if I even did the first step properly. Can any show me how to solve using this method?
Best Answer
Follow the following thirteen steps, Rx (like R1, R2 or R3) refers to the matrix row number:
You should arrive at a RREF of:
$$ \left[ \begin{array}{@{}ccc|c@{}}1 & 0 & 0 & 3\\0 & 1 & 0 & 1\\0 & 0 & 1 & 2\end{array}\right] $$ Now, you read the solution from the bottom up, so we have:
You should of course check my work and verify that those value satisfy all three simultaneous equations!