Expert: Richard J. Raridon - 10/23/2007

Question
i dont understand how to solve this equation please help??
2w-x+5y+z=-3
3w+2x+2y-6z=-32
w+3x+3y-z=-47
5w-2x-3y+3z=49

Answer
One way to solve them is by using determinants.  First you create a determinant, D, of the coefficients of the unknowns, thus
|2 -1 5 1 |
|3 2 2 -6 |
|1 3 3 -1 |
|5 -2 -3 3|
and solve it.  
To find w, create another determinant, W, by replacing the first column with the numbers on the other side of the equals signs, thus
|-3 -1 5 1 |
|-32 2 2 -6|
|-47 3 3 -1|
|49 -2 -3 3|
evaluate that and then w = W/D
Repeat for the 2nd column to calculate x, the 3rd column to calculate y and the 4th column to calculate z.  
The other method is similar to what you would do for two equations, two unknowns, namely, eliminate one of the variables.  So here you would pick some variable, such as z.  By adding eq. 1 and eq. 3, you eliminate z.  Multiply eq. 1 by 6 and add it to eq. 2 to eliminate z. Multiply eq.1 by 3 and subtract it from eq. 4 to eliminate z.  
Now you just have w, x, and y.  Repeat the process to eliminate one of those, etc.  Either way is tedious and you have to be careful not to make errors.