# How do you solve the system x +4y - 6z = 4, 4x - 3y + 2z = 26, x - 2y +2z = 8?

Nov 12, 2016

Please see the explanation for step leading to the solution:

$x = 4 , y = - 6 , z = - 4$

#### Explanation:

Write the 3 equations as an augmented matrix:

[ (1,4,-6,|,4), (4,-3,2,|,26), (1,-2,2,|,8) ]

Multiply row 1 by -4 and add to row 2:

[ (1,4,-6,|,4), (0,-19,26,|,10), (1,-2,2,|,8) ]

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

[ (1,4,-6,|,4), (0,-19,26,|,10), (0,-6,8,|,4) ]

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

[ (1,4,-6,|,4), (0,-1,2,|,-2), (0,-6,8,|,4) ]

Multiply row 2 by -1:

[ (1,4,-6,|,4), (0,1,-2,|,2), (0,-6,8,|,4) ]

Multiply row 2 by 6 and add to row 3:

[ (1,4,-6,|,4), (0,1,-2,|,2), (0,0,-4,|,16) ]

Divide row 3 by -4:

[ (1,4,-6,|,4), (0,1,-2,|,2), (0,0,1,|,-4) ]

Multiply row 3 by 2 and add to row 2:

[ (1,4,-6,|,4), (0,1,0,|,-6), (0,0,1,|,-4) ]

Multiply row 3 by 6 and add to row 1:

[ (1,4,0,|,-20), (0,1,0,|,-6), (0,0,1,|,-4) ]

Multiply row 2 by -4 and add to row 1:

[ (1,0,0,|,4), (0,1,0,|,-6), (0,0,1,|,-4) ]

This means that $x = 4 , y = - 6 , \mathmr{and} z = - 4$
You should check my answer by substution into the orignal equations.