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Question 182659: I have to do the Gauss Method for:
x+y+z=2
2x-3y+2z=4
4x+y-3z=1
My teacher doesn't require us to use a textbook.
Answer by MathGuyJoe(20)   (Show Source):
You can put this solution on YOUR website! Your first step is to write out the coefficient matrix:
You want to end up with a matrix that looks like this (triangular form):
To do that, you perform a combination of the elementary row operations:
1) Switch any 2 rows
2) Multiply each row element by a non-zero constant
3) Replace a row by adding its values to a multiple of another row
Since our first row is all 1's, it's easy to pick multipliers that will 'zero' out the required fields in rows 2 & 3 when we add the rows together (using row operation 3). So let's replace row 2 with (-2 * Row 1) + Row 2 -- here's the shorthand way to say that:
So basically, Row 1 and Row 3 remain the same, only row 2 changes:
So that got us a zero under the 1 in the first column which is why we picked -2 as the multiplier. Next, we'll replace row 3 with (-4 * Row 1) + Row 3:
So now, our 1st column looks like we want it. We only have one more zero to create (in the 3rd row, 2nd column), so we'll choose  as our multiplier and use it against row 2 this time:
We now have our matrix in triangular form -- let's write the equations using the coefficients from it:
So solving from the bottom up (reverse substitution), we know z = 1 and y = 0. Substituting these values into the top equation gives us x = 1. As a final check, you can substitute these values in each of the 3 original equations:
Hope this helps. ~ Joe
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