Question

Solve the system by back substitution.

-2x - 4y - z - 3w = -5
y + 6z + 2w = 8
4z + 4w = 8
-4w = -12
Solution should be set like {( _ , _ , _ , _ )}

Answer 1

w = 3, x = -35/2, y = 8, z = -1

Explanation:

Solve the following system:

{-3 w - 2 x - 4 y - z = -5

{2 w + y + 6 z = 8

{4 z + 4 w = 8

{-4 w = -12

In the fourth equation, look to solve for w:

{-3 w - 2 x - 4 y - z = -5

{2 w + y + 6 z = 8

{4 z + 4 w = 8

{-4 w = -12

Divide both sides by -4:

{-3 w - 2 x - 4 y - z = -5

{2 w + y + 6 z = 8

{4 z + 4 w = 8

{w = 3

Substitute w = 3 into the first, second, and third equations:

{-9 - 2 x - 4 y - z = -5

{6 z + y + 6 = 8

{4 z + 12 = 8

{w = 3

In the third equation, look to solve for z:

{-9 - 2 x - 4 y - z = -5

{6 z + y + 6 = 8

{4 z + 12 = 8

{w = 3

Subtract 12 from both sides:

{-9 - 2 x - 4 y - z = -5

{6 z + y + 6 = 8

{4 z = -4

{w = 3

Divide both sides by 4:

{-9 - 2 x - 4 y - z = -5

{6 z + y + 6 = 8

{z = -1

{w = 3

Substitute z = -1 into the first and second equations:

{-4 y - 2 x - 8 = -5

{y = 8

{z = -1

{w = 3

Substitute y = 8 into the first equation:

{-2 x - 40 = -5

{y = 8

{z = -1

{w = 3

In the first equation, look to solve for x:

{-2 x - 40 = -5

{y = 8

{z = -1

{w = 3

Add 40 to both sides:

{-2 x = 35

{y = 8

{z = -1

{w = 3

Divide both sides by -2:

{x = -35/2

{y = 8

{z = -1

{w = 3

Collect results in alphabetical order:

Answer: {w = 3, x = -35/2, y = 8, z = -1