# How do you solve the system x-2y+2z=-2, -x+4y-2z=3, 2x-4y+z=-7?

##### 1 Answer
Nov 15, 2016

Please see the explanation for steps leading to the solution:

$x = - 3 , y = \frac{1}{2} , \mathmr{and} z = 1$

#### Explanation:

Write the 3 equations as an augmented matrix:

[ (1,-2,2,|,-2), (-1,4,-2,|,3), (2,-4,1,|,-7) ]

Add row 1 to row 2:

[ (1,-2,2,|,-2), (0,2,0,|,1), (2,-4,1,|,-7) ]

Multiply row 1 by -2 and add to row 3:

[ (1,-2,2,|,-2), (0,2,0,|,1), (0,0,-3,|,-3) ]

Divide row 3 by -3:

[ (1,-2,2,|,-2), (0,2,0,|,1), (0,0,1,|,1) ]

Add row 2 to row 1:

[ (1,0,2,|,-1), (0,2,0,|,1), (0,0,1,|,1) ]

Divide row 2 by 2:

[ (1,0,2,|,-1), (0,1,0,|,1/2), (0,0,1,|,1) ]

Multiply row 3 by -2 and add to row 1:

[ (1,0,0,|,-3), (0,1,0,|,1/2), (0,0,1,|,1) ]

The means that $x = - 3 , y = \frac{1}{2} , \mathmr{and} z = 1$

check:

$\left(- 3\right) - 2 \left(\frac{1}{2}\right) + 2 \left(1\right) = - 2$
$- \left(- 3\right) + 4 \left(\frac{1}{2}\right) - 2 \left(1\right) = 3$
$2 \left(- 3\right) - 4 \left(\frac{1}{2}\right) + 1 = - 7$

$- 2 = - 2$
$3 = 3$
$- 7 = - 7$

This checks