Gaussian elimination aims to orient the matrix formed by the system of equations such that only zeros appear below the pivot points of each row:
After performing Gaussian elimination, you are left with the following system of equations:
$x\+y\+z\=3\phantom{\rule[0.0ex]{0.0em}{0.0ex}}yplus;5zequals;6\phantom{\rule[0.0ex]{0.0em}{0.0ex}}13zequals;26$
You can then solve for z:
$z\=\frac{26}{13}$
$z\=2$
and substitute it into the second equation:
$y\+5\left(2\right)\=6$
$y\=6\+10$
$y\=4$
Finally, you can substitute the y and z into the first equation to solve for x:
$x\+42\=3$
$x\=32$
$x\=1$
Therefore, the solution to the equation is:
$x\=1yequals;4zequals;2$
or:
$\left(1\,4\,2\right)$
