4-variable system?

Question

1)5x-2y-z+2w=-44
2)3x+3y-4z+w=-27
3)-2x+4y+5z-2w=48
4)4x-5y+6z+4w=-23

now i tried 1 and 3, 2 and 3, 5 and 6 and none fit into the equations.

Answer 1

Doing blind Gaussian elimination will result in a lot of work. Here is a solution that still requires a lot of work, but less than a direct approach. First, set this up as a matrix:

[5, -2, -1, 2 | -44]
[3, 3, -4, 1 | -27]
[-2, 4, 5, -2 | 48]
[4, -5, 6, 4 | -23]

Swap rows 1 and 2:

[3, 3, -4, 1 | -27]
[5, -2, -1, 2 | -44]
[-2, 4, 5, -2 | 48]
[4, -5, 6, 4 | -23]

Now, subtract 2*row 1 from row 2, add 2*row 1 to row 3, and subtract 4*row 1 from row 4 (in this case, we are clearing out the column on the right):

[3, 3, -4, 1 | -27]
[-1, -8, 7, 0 | 10]
[4, 10, -3, 0 | -6]
[-8, -17, 22, 0 | 85]

Now, since we don't want to do any divisions, we'll clear out the left column next. Add 3*row 2 to row 1, add 4*row 2 to row 3, subtract 8*row 2 from row 4, and then multiply row 2 by -1. So we have:

[0, -21, 17, 1 | 3]
[1, 8, -7, 0 | -10]
[0, -22, 25, 0 | 34]
[0, 47, -34, 0 | 5]

Okay, now at this point, there's really no further eliminations we can make that won't require working with some very ugly fractions. But notice that the equations for y and z no longer depend on x or w, so we can break this off and solve it as a subsystem:

[-22, 25][y] . [34]
[47, -34][z] = [5]

The determinant of this matrix is -22*(-34) - 47*25 = -427. Replacing the left column with the column vector (34, 5), we obtain:

[34, 25]
[5, -34]

With determinant 34*(-34) - 5*25 = -1281. So by Cramer's rule, y = -1281/-427 = 3. And replacing the right column with (34, 5), we obtain:

[-22, 34]
[47, 5]

With determinant -22*5 - 47*34 = -1708. So z = -1708/-427 = 4. So now returning to the first matrix:

[0, -21, 17, 1 | 3]
[1, 8, -7, 0 | -10]
[0, -22, 25, 0 | 34]
[0, 47, -34, 0 | 5]

We see the equations governing x and w are:

x + 8y - 7z = -10, and
-21y + 17z + w = 3.

Substituting the values we found for y and z:

x + 8(3) - 7(4) = -10
-21(3) + 17(4) + w = 3

Simplifying:

x - 4 = -10
5 + w = 3

x=-6, w=-2

So our final solution is x=-6, y=3, z=4, and w=-2

Answer 2

have you tried solving using a matrix?