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 Question from Jennifer, a student: Hello, my name is Jennifer. I am a 12th grader in Pre-Calculus and I was wondering if you may help me with Row Echelon Form. x + 2y - 3z = -5 -2x - 4y - 6z = 10 3x + 7y - 2z = -13

Hi Jennifer,

The augmented matrix for this linear system is

$\left( \begin{array}{rrrr} 1 & 2 & -3 & -5\\ -2 & -4 & -6 & 10\\ 3 & 7 & -2 & -13 \end{array} \right)$

To make the first entry in the second row equal to $0$ I would multiply the first row by 2 and add to the second row. The result is

$\left( \begin{array}{rrrr} 1 & 2 & -3 & -5\\ 0 & 0 & -12 & 0\\ 3 & 7 & -2 & -13 \end{array} \right)$

Wow!

Multiply the second row by $\large \frac{-1}{12}$ and the matrix becomes

$\left( \begin{array}{rrrr} 1 & 2 & -3 & -5\\ 0 & 0 & 1 & 0\\ 3 & 7 & -2 & -13 \end{array}. \right)$

The second row is in the form you want for the third row so interchange the second and third row to get

$\left( \begin{array}{rrrr} 1 & 2 & -3 & -5\\ 3 & 7 & -2 & -13\\ 0 & 0 & 1 & 0 \end{array}. \right)$

Can you see what to do now?

Penny

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