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Question 336153: How do I solve this problem. I know it's done through matrices but I'm still very confused on the concept so explanation would be greatly appreciated.
3x-y+z= -9
2x+y-z= -1
x-y-z= -7
Answer by tinbar(133)   (Show Source):
You can put this solution on YOUR website! first u have to make this matrix, this is done by taking the coeffecients of the variables and solutions
[3 -1 1|-9]
[2 1 -1|-1]
[1 -2 -2|-7]
the first column contains the number 3, 2, and 1, we want to somehow turn the 2 and 1 into a 0 (this is equivalent to saying let's get rid of the x variable in the first and second equations)
if we multiply the first row by 2 and subtract it from 3 times the second row, this will get rid of the two. firstly, row 1 is not going to be changed so we can just rewrite that and the second row will be replaced by the operation we just figured out (2*r1-3*r2)->r2....(operation)->row being replaced by operation
[3 -1 1|-9]
[0 -5 5|-15]
[1 -2 -2|-7]
to get rid of the x in row 3 we can do (r1-3*r3)->row3
[3 -1 1|-9]
[0 -5 5|-15]
[0 5 7| 12]
now we have to do the same thing as we did for the x, to the y and this time we won't start at the same row, we will move down one. (in other words, when we got rid of x, we left x alone in row 1 and only removed it from row 2 and 3, for y, we r going to leave alone rows 1 and 2 and only mess with row 3
the operation we could do is (r2+r3)->r3
[3 -1 1|-9]
[0 -5 5|-15]
[0 0 12|-3]
what have is called row-echelon form, we can go a step further and make it reduced row echelon form, either we have the answer. from this matrix, the last row equivalently now says 12z=-3, so we can get z, from that we can replace z in row 2 and get y, after that we can replace z and y in row 1 to get x
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