Math CentralQuandaries & Queries


Question from Johan, a student:

Hi, Im' Johan from Malaysia and I'm doing my first year engineering degree. I need some help in solving this question or maybe a proper method of approaching gaussian elimination.
Your help is appreciated. Thanks!

x + 2y - 3z + 4w = 12
2x + 2y - 2z + 3w = 10
0 + y + z + 0 = -1
x - y + z - 2w = -4

Hi Johan.

Once a professor taught me a very important rule: When you have n unknowns, you need at least n equations to solve for all of them.

You have four equations and four unknowns, so I expect that you'll be able to find the solution using regular "simultaneous equation" solving methods, such as substitution and elimination.

The idea is to combine the equations in order to reduce the number of variables.

Here's an example with "elimination": I decide I want to eliminate the "w", so I look at the four equations and I choose two where I can see that the factor of w in one is an easy multiply of the factor of w in the other, then I multiply one of the equations in order to make the factors identical.

The top and bottom equations look right: the top has 4w and the bottom has -2w, so if I multiply the bottom equation (both sides!) by -2, I get a new fourth equation:

-2x + 2y - 2z + 4w = 8

Now I can subtract the first equation. I will subtract the left side of the first equation from the left side of the equation above and will subtract the right sides similarly:

(-2x + 2y - 2z + 4w) - (x + 2y - 3z +4w) = (8) - (12)

-3x + z = -4.

As an unexpected bonus, this also eliminated the y from the equation.

Now I need to combine a different pair of equations and come up with another equation without w in it. I'll look at the second and fourth equations. Because I have +3w and -2w, I will multiply the second equation by 2 and the bottom equation by -3, so both will be +6w:

2( 2x + 2y - 2z + 3w) - (-3)(x - y + z - 2w)= 2(10) - (-3)(-4)
4x + 4y - 4z + 6w + 3x - 3y + 3z - 6w = 20 - 12

7x + y - z = 8.

Notice that the third equation has no w in it. So we now have these three equations in three unknowns:

7x + y - z = 8,
-3x + z = -4,
y + z = -1.

Now we do the same kind of thing: get rid of one of the variables and come up with two different equations in two variables. Since the bottom equation has no x in it already, let's combine the first and second equations in such a way that we can remove the x. This time I'll show you the "substitution" method.

First we solve one of the equations for the variable we want to eliminate (x):

-3x + z = -4
3x = z + 4

x = z/3 + 4/3.

Now we "substitute" the right side of this into the other equation (the 7x + y - z = 8):

7x + y - z = 8
7(z/3 + 4/3) + y - z = 8
z(7/3 - 1) + y = 8 - 7(4/3)
y + (4/3)z = -4/3

3y + 4z = -4.

So now we have two equations with two unknowns:

3y + 4z = -4,
y + z = -1.

Now you can use again either the elimination method or the substitution method to solve for either y or z. With that value, you can plug the value into one of the above equations to get the other value. Then you have the values of y and z, so you can plug them into one of the equations with three variables and get the value of z, then do the same to one of the original equations to find w.

Stephen La Rocque.

About Math Central


Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.
Quandaries & Queries page Home page University of Regina PIMS