`3 w+x = 4` (equation 1) `-w+2 y-z = 0 ` (equation 2) `3 y-2 w = 1 ` (equation 3) `2 x-y+4 z = 5 ` (equation 4)

Add 1/3 × (equation 1) to equation 2:

`3 w+x+0 y+0 z = 4` (equation 1) `0 w+x/3+2 y-z = 4/3 ` (equation 2) `-(2 w)+0 x+3 y+0 z = 1 ` (equation 3) `0 w+2 x-y+4 z = 5 ` (equation 4)

Multiply equation 2 by 3:

`3 w+x+0 y+0 z = 4 ` (equation 1) `0 w+x+6 y-3 z = 4` (equation 2) `-(2 w)+0 x+3 y+0 z = 1` (equation 3) `0 w+2 x-y+4 z = 5` (equation 4)

Add 2/3 × (equation 1) to equation 3:

`3 w+x+0 y+0 z = 4` (equation 1) `0 w+x+6 y-3 z = 4` (equation 2) `0 w+(2 x)/3+3 y+0 z = 11/3` (equation 3) `0 w+2 x-y+4 z = 5` (equation 4)

Multiply equation 3 by 3:

`3 w+x+0 y+0 z = 4` (equation 1) `0 w+x+6 y-3 z = 4` (equation 2) `0 w+2 x+9 y+0 z = 11` (equation 3) `0 w+2 x-y+4 z = 5` (equation 4)

Swap equation 2 with equation 3:

`3 w+x+0 y+0 z = 4` (equation 1) `0 w+2 x+9 y+0 z = 11` (equation 2) `0 w+x+6 y-3 z = 4 ` (equation 3) `0 w+2 x-y+4 z = 5` (equation 4)

Subtract 1/2 × (equation 2) from equation 3:

`3 w+x+0 y+0 z = 4 ` (equation 1) `0 w+2 x+9 y+0 z = 11 ` (equation 2) `0 w+0 x+(3 y)/2-3 z = (-3)/2` (equation 3) `0 w+2 x-y+4 z = 5` (equation 4)

Multiply equation 3 by 2/3:

` 3 w+x+0 y+0 z = 4 ` (equation 1) `0 w+2 x+9 y+0 z = 11 ` (equation 2) `0 w+0 x+y-2 z = -1 ` (equation 3) `0 w+2 x-y+4 z = 5 ` (equation 4)

Subtract equation 2 from equation 4:

`3 w+x+0 y+0 z = 4 ` (equation 1) `0 w+2 x+9 y+0 z = 11` (equation 2) `0 w+0 x+y-2 z = -1` (equation 3) `0 w+0 x-10 y+4 z = -6 ` (equation 4)

Divide equation 4 by 2:

`3 w+x+0 y+0 z = 4` (equation 1) `0 w+2 x+9 y+0 z = 11 ` (equation 2) `0 w+0 x+y-2 z = -1 ` (equation 3) `0 w+0 x-5 y+2 z = -3 ` (equation 4)

Swap equation 3 with equation 4:

`3 w+x+0 y+0 z = 4 ` (equation 1) `0 w+2 x+9 y+0 z = 11 ` (equation 2) `0 w+0 x-5 y+2 z = -3 ` (equation 3) `0 w+0 x+y-2 z = -1` (equation 4)

Add 1/5 × (equation 3) to equation 4:

`3 w+x+0 y+0 z = 4 ` (equation 1) `0 w+2 x+9 y+0 z = 11 ` (equation 2) `0 w+0 x-5 y+2 z = -3 ` (equation 3) `0 w+0 x+0 y-(8 z)/5 = (-8)/5 ` (equation 4)

Multiply equation 4 by -5/8:

`3 w+x+0 y+0 z = 4` (equation 1) `0 w+2 x+9 y+0 z = 11` (equation 2) `0 w+0 x-5 y+2 z = -3` (equation 3) `0 w+0 x+0 y+z = 1 ` (equation 4)

Subtract 2 × (equation 4) from equation 3:

`3 w+x+0 y+0 z = 4` (equation 1) `0 w+2 x+9 y+0 z = 11` (equation 2) `0 w+0 x-5 y+0 z = -5` (equation 3) `0 w+0 x+0 y+z = 1` (equation 4)

Divide equation 3 by -5:

`3 w+x+0 y+0 z = 4 ` (equation 1) `0 w+2 x+9 y+0 z = 11` (equation 2) `0 w+0 x+y+0 z = 1` (equation 3) `0 w+0 x+0 y+z = 1` (equation 4)

Subtract 9 × (equation 3) from equation 2:

`3 w+x+0 y+0 z = 4 ` (equation 1) `0 w+2 x+0 y+0 z = 2 ` (equation 2) `0 w+0 x+y+0 z = 1 ` (equation 3) `0 w+0 x+0 y+z = 1 ` (equation 4)

Divide equation 2 by 2:

`3 w+x+0 y+0 z = 4 ` (equation 1) `0 w+x+0 y+0 z = 1` (equation 2) `0 w+0 x+y+0 z = 1` (equation 3) 0 w+0 x+0 y+z = 1 (equation 4)

Subtract equation 2 from equation 1:

`3 w+0 x+0 y+0 z = 3 ` (equation 1) `0 w+x+0 y+0 z = 1 ` (equation 2) `0 w+0 x+y+0 z = 1 ` (equation 3) `0 w+0 x+0 y+z = 1 ` (equation 4)

Divide equation 1 by 3:

`w+0 x+0 y+0 z = 1 ` (equation 1) `0 w+x+0 y+0 z = 1 ` (equation 2) `0 w+0 x+y+0 z = 1 ` (equation 3) `0 w+0 x+0 y+z = 1 ` (equation 4) w = 1 x = 1 y = 1 z = 1

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