How do you solve $x + 4 y - z = 7$ $x+4y-z=7$ , $x + y + z = 2$ $x+y+z=2$ and $- 2 x - 2 y + 2 z = - 8$ $-2x-2y+2z=-8$ using matrices?

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Answer 1 · 2016-10-14T04:38:07

Please see the explanation for the process.
$x = 2, y = 1, z = - 1$ $x = 2, y = 1, z = -1$

Write $x + 4 y - z = 7$ $x + 4y - z = 7$ into the first row of an augmented matrix:

$[ (1,4,-1,|,7) ]$

Write $x + y + z = 2$ into the second row:

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

Write $-2x - 2y + 2z = -8$ into the third row:

$[ (1,4,-1,|,7), (1,1,1,|,2), (-2, -2, 2, |, -8) ]$

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

$[ (1,4,-1,|,7), (0,-3,2,|,-5), (-2, -2, 2, |, -8) ]$

Multiply row 1 by 2 and add to row 3:

$[ (1,4,-1,|,7), (0,-3,2,|,-5), (0, 6, 0, |, 6) ]$

Divide row 3 by 6 and swap with row 2:

$[ (1,4,-1,|,7), (0, 1, 0, |, 1), (0,-3,2,|,-5) ]$

Multiply row 2 by 3 and add to row 3:

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

Divide row 3 by 2:

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

Add row 3 to row 1:

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

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

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

$x = 2, y = 1, z = -1$

How do you solve x+4y-z=7x+4y−z=7x+4y-z=7, x+y+z=2x+y+z=2x+y+z=2 and -2x-2y+2z=-8−2x−2y+2z=−8-2x-2y+2z=-8 using matrices?