SOLUTION: use goussian reduction method to solve the following linear system 2x-y+4z=1 y+z=3

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Question 1090283: use goussian reduction method to solve the following linear system
2x-y+4z=1
y+z=3
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!

There are only two equations and three unknowns. So if the system has a solution, it will be an infinite set of solutions defined by a parameter.

Because of the way Gaussian reduction is performed, the solution will have x and y expressed in terms of parameter z.


The matrix for the given system of equations is
matrix%282%2C4%2C2%2C-1%2C4%2C1%2C0%2C1%2C1%2C3%29

I don't like introducing fractions into the matrix when doing Gaussian reduction; but in this case we have no choice. So divide the first row by 2:

matrix%282%2C4%2C1%2C-1%2F2%2C2%2C1%2F2%2C0%2C1%2C1%2C3%29

Next use the 1 in row 2 column 2 to get a 0 in row 1 column 2:

R1 <-- R1 + (1/2)R2: 1+0=1; -1/2+1/2=0; 2+1/2=5/2; 1/2+3/2=2.
matrix%282%2C4%2C1%2C0%2C5%2F2%2C2%2C0%2C1%2C1%2C3%29

This is as far as we can go with Gaussian reduction. The final matrix gives us these equations:

x+(5/2)z=2; y+z=3

We rewrite these equations to give us parametric equations for x and y in terms of parameter z:

x = 2-(5/2)z; y = 3-z

And the parametric solution set is

x = 2-(5/2)z;
y=3-z;
z=z