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Algebra/Simultaneous linear equation with three unknowns

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Question
Solving simultaneous linear equation with three unknows is a damn knotty problem to me. Though, with the help of determinant, I can solve it effectively. But now, I don't wanna use determinants  anymore.  Thus, I've tried an alternative however to no avail.  Pls. Prof.  Treat me like a novice by solving the question below without using inverse matrix or determinants:


:( 3x+2y-z=19.........(1)
       4x-y+2z=4...........(2)
        2x+4y-5z=32....(3)



:(5x-3y-2z=31..........(1)
    2x+6y+3z=4..........(2)
    4x+2y-z=30............(3)
 

explain thoroughly how you solve it!

Answer
To solve each of them, I write down the problem in matrix format.
Once this has been done, I get a 1 in the 1st element of the 1st by dividing that row by 3.
I then get a zero in each cell below that by subtracting off the appropriate multiple of row 1.

For the 1st step, I divide row 1 by 3, subtract 4/3 for the 2nd row, and subtract 2/3 from the 3rd row.  I also do it to the solutions.  This gives
1   0.6667  -0.3333     6.3333
0  -3.6667   3.3333   -21.3333
0   2.6667  -4.3333    19.3333

In the next step, I multiply the 2nd row by -3/11.  This puts a 1 in the place of -3.6667.
I multiply the 2nd row by 1/11 and add that to the 1st row.  I multiply the 2nd row by 8/11 and add it the 3rd row.  This gives a zero in the second column.  The result is
1  0   0.2727  2.4545
0  1  -0.9091  5.8182
0  0  -1.9091  3.8182

To get a 1 in the 3rd position of the 3rd row, multiply that row by -11/21.
Add -10/21 of the 3rd row to the 2nd row.  Add -1/7 of the 3rd row to the 1st row.

The solution turns out to be 3, 4, -2.

This is how to do the 2nd problem as well.  

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Scott A Wilson

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Any algebraic question you've got. That includes question that are linear, quadratic, exponential, etc.

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I have solved story problems, linear equations, parabolic equations. I have also solved some 3rd order equations and equations with multiple variables.

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MS at math OSU in mathematics at OSU, 1986. BS at OSU in mathematical sciences (math, statistics, computer science), 1984.

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