document.write( "Question 1090283: use goussian reduction method to solve the following linear system
\n" ); document.write( "2x-y+4z=1
\n" ); document.write( "y+z=3 \n" ); document.write( "
Algebra.Com's Answer #704902 by greenestamps(13200)\"\" \"About 
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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.

\n" ); document.write( "Because of the way Gaussian reduction is performed, the solution will have x and y expressed in terms of parameter z.


\n" ); document.write( "The matrix for the given system of equations is
\n" ); document.write( "\"matrix%282%2C4%2C2%2C-1%2C4%2C1%2C0%2C1%2C1%2C3%29\"

\n" ); document.write( "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:

\n" ); document.write( "\"matrix%282%2C4%2C1%2C-1%2F2%2C2%2C1%2F2%2C0%2C1%2C1%2C3%29\"

\n" ); document.write( "Next use the 1 in row 2 column 2 to get a 0 in row 1 column 2:

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

\n" ); document.write( "This is as far as we can go with Gaussian reduction. The final matrix gives us these equations:

\n" ); document.write( "x+(5/2)z=2; y+z=3

\n" ); document.write( "We rewrite these equations to give us parametric equations for x and y in terms of parameter z:

\n" ); document.write( "x = 2-(5/2)z; y = 3-z

\n" ); document.write( "And the parametric solution set is

\n" ); document.write( "x = 2-(5/2)z;
\n" ); document.write( "y=3-z;
\n" ); document.write( "z=z\r
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