Algebraic equations in four variables.?

Question

W + X + Y + Z = 2

W + 2X + 2Y + 4Z = 1

W - X + Y + Z = 6

W - 3X - Y + Z = 2

I need to know what numbers represent the variables above. Please explain in as detailed a way as possible, how you arrived at the numbers, whether by using a calculator, or by hand. I have a TI-86, so if you are familiar with that calc, please tell me how to solve this, in as detailed a manner as possible, using a 4 x 4 matrix.

I am more or less an Algebra newb, so please don't be afraid to break this down into as simple a form as possible. Thanks.

Answer 1

Subtract equation 1 from equation 3.

-2x = 4 ~> x = -2

Rewrite equation 1 and equation 4 with this new information.

W - 2 + Y + Z = 2

W + 6 - Y + Z = 2

Move constants to the other side.

W + Y + Z = 4

W - Y + Z = -4

Subtract the bottom equation from the top

2Y = 8

Y = 4

Substitute both of these values into equation 1

W - 2 + 4 + Z = 2 ~> W + Z = 0

Substitute known values of X and Y into equation 2

W - 4 + 8 + 4Z = 1 ~> W + 4Z = -3

From the rewrite of equation 1, Z = -W

Substitute into the rewrite of equation 2

W - 4W = -3
-3W = -3
W = 1

Since we know W and Z are opposites, Z = -1

W = 1, X = -2, Y = 4 and Z = -1

Answer 2

These are simultaneous equations

The first step is to label then (A,B,C and D)

Now the idea with solving simultaneous equations is to add and subtract (also multiply and divide) the different equations together in order to eliminate the different variables (X,Y,Z and X)

For example in this case
A-B gives (1st minus the 3rd)

2X = -4

which eliminates W, Y and Z and gives you the value of X

A more general procedure is called gaussian elimination but I think this one can be done by hand.