# Solving Systems by Gauss-Jordan Elimination

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?

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If equation 1 is correct the augmented matrix is \begin{align} \begin{bmatrix} 1 & 1 & 2 & 8\\ -1 & -2 & 3 & 1\\ 3 & -7 & 4 & 10 \end{bmatrix} \end{align} –  Américo Tavares May 12 '13 at 18:50
Why would you subtract 1 from $x_2$ and $x_3$ in the first row? –  user1327636 May 12 '13 at 19:17

Follow the following thirteen steps, Rx (like R1, R2 or R3) refers to the matrix row number:

• Swap R1 with R3
• Multiply R2 by -3/13
• Subtract R1/3 from R3
• Multiply (3/2)R3
• Swap R2 with R3
• Subtract R2/5 from R3
• Multiply (-5/6)R3
• Subtract R3 from R2
• Subtract (4) R3 from R1
• Add (7/5) R2 to R1
• Divide R1 by 3
• Divide R2 by 5

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:

• $x_3 = 2$
• $x_2 = 1$
• $x_1 = 3$

You should of course check my work and verify that those value satisfy all three simultaneous equations!

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Helpful solution/algorithm/direction! $\ddot\smile$ –  amWhy May 13 '13 at 0:19
I think the OP got it and sometimes they need a little nudge off the cliff sort of speak! :-) –  Amzoti May 13 '13 at 0:21