# How do you solve using gaussian elimination or gauss-jordan elimination, 3x + y + 2z = 3, 2x - 37 - z = -3, x + 2y + z = 4?

Jan 19, 2016

$x = \frac{617}{276}$
$y = - \frac{15}{92}$
$z = \frac{577}{276}$

#### Explanation:

I assume that there is a transcription error in the question and the second term in the second equation should actually be a $y$ term.

$3 x + y + 2 z = 3$
$2 x - 37 y - z = - 3$
$x + 2 y + z = 4$

Adding together the second and third equations gives us
$3 x - 35 y = 1$

Next, doubling the second equation and adding it to the first gives
$7 x - 73 y = - 3$

Solving these two equations gives $7 \cdot \frac{3 + 35 y}{3} - 73 y = - 3$
$21 + 35 y - 219 y \equiv - 9$
$184 y = - 30$
$y = - \frac{15}{92}$

$\therefore 3 x = 1 + 35 \cdot \frac{15}{92} = \frac{92 + 525}{92} = \frac{617}{92}$
$x = \frac{617}{276}$

Then $z = 4 - \frac{617}{276} - 2 \cdot \left(- \frac{15}{92}\right)$
$z = \frac{1104 - 617 + 90}{276} = \frac{577}{276}$