# `x + 3w = 4, 2y - z - w = 0, 3y - 2w = 1, 2x - y + 4z = 5` Solve the system of linear equations and check any solutions algebraically.

````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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