# Solving elementary row operations

So I am faced with the following: \begin{align} x_1 + 4x_2 - 2x_3 +8x_4 &=12\\ x_2 - 7x_3 +2x_4 &=-4\\ 5x_3 -x_4 &=7\\ x_3 +3x_4 &=-5 \end{align}

How should I approach this problem? In other words, what is the next elementary row operation that I should attempt in order to solve it? I know how to do with 3 equations by using the augmented method but this got me a little confused.

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Are you familiar with the method of Gaussian elimination? – Alex Wertheim Jun 5 '13 at 4:42

Write it as an augmented system:

$$\left[\begin{array}{cccc|c} 1& 4& -2& 8 & 12\\ 0& 1& -7& 2 & -4\\ 0& 0& 5& -1 & 7\\ 0& 0& 1& 3 & -5 \end{array}\right]$$

Gaussian Elimination (Row-Reduced-Echelon-Form - RREF) will yield:

$$\left[\begin{array}{cccc|c} 1& 0& 0& 0 & 2\\ 0& 1& 0& 0 & 7\\ 0& 0& 1 & 0 & 1\\ 0& 0& 0& 1 & -2 \end{array}\right]$$

Thus:

• $x_4 = -2$
• $x_3 = 1$
• $x_2 = 7$
• $x_1 = 2$
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Putting matrices into the Latex area has been difficult for me, but it has been easy for you like drinking a glass of water. :D – Babak S. Jun 5 '13 at 7:14
@BabakS.: I created a little text file cheat sheet that I use for common LaTex items that I reuse because I am still too slow at typing things up and some of you are very fast! Feels like a time trial at times. Hope all is well with you my friend! Have a great day! – Amzoti Jun 5 '13 at 12:16
I wish I could have that valuable cheat sheet. Wanna sell it? How many $?? :D – Babak S. Jun 5 '13 at 13:14 Cheat sheet to the rescue! +1 – amWhy Jun 6 '13 at 0:21 HINT: Use Elimination/ Substitution or Cross Multiplication to solve for$x_3,x_4$from the last two simultaneous equation. Putting the values of$x_3,x_4$in the second equation, you will get$x_2$Putting the values of$x_2,x_3,x_4$in the first equation you will get$x_1$- By the Gauss-Jordan algorithm, your next step would be to make the$4x_{2}$into a$0\$, by the Type 3 elementary row operation: multiply row 2 by -4 and add that to row 1.

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