document.write( "Question 182659: I have to do the Gauss Method for:\r
\n" ); document.write( "\n" ); document.write( "x+y+z=2
\n" ); document.write( "2x-3y+2z=4
\n" ); document.write( "4x+y-3z=1\r
\n" ); document.write( "\n" ); document.write( "My teacher doesn't require us to use a textbook. \n" ); document.write( "
Algebra.Com's Answer #137144 by MathGuyJoe(20)\"\" \"About 
You can put this solution on YOUR website!
Your first step is to write out the coefficient matrix:
\n" ); document.write( "\"+matrix%283%2C4%2C+1%2C1%2C1%2C2%2C+2%2C-3%2C2%2C4%2C+4%2C+1%2C+-3%2C+1%29+\"
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\n" ); document.write( "You want to end up with a matrix that looks like this (triangular form):
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\n" ); document.write( "To do that, you perform a combination of the elementary row operations:
\n" ); document.write( "1) Switch any 2 rows
\n" ); document.write( "2) Multiply each row element by a non-zero constant
\n" ); document.write( "3) Replace a row by adding its values to a multiple of another row
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\n" ); document.write( "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:
\n" ); document.write( "\"+-2R%5B1%5D+%2B+R%5B2%5D+-%3E+R%5B2%5D+\"
\n" ); document.write( "So basically, Row 1 and Row 3 remain the same, only row 2 changes:
\n" ); document.write( "\"+matrix%283%2C4%2C+1%2C+1%2C+1%2C+2%2C+0%2C+-5%2C+0%2C+0%2C+4%2C+1%2C+-3%2C+1%29+\"
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\n" ); document.write( "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:
\n" ); document.write( "\"+-4R%5B1%5D+%2B+R%5B3%5D+-%3E+R%5B3%5D+\"
\n" ); document.write( "\"+matrix%283%2C4%2C+1%2C+1%2C+1%2C+2%2C+0%2C+-5%2C+0%2C+0%2C+0%2C+-3%2C+-7%2C+-7%29+\"
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\n" ); document.write( "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 \"+-3%2F5+\" as our multiplier and use it against row 2 this time:
\n" ); document.write( "\"+%28-3%2F5%29R%5B2%5D+%2B+R%5B3%5D+-%3E+R%5B3%5D+\"
\n" ); document.write( "\"+matrix%283%2C4%2C+1%2C+1%2C+1%2C+2%2C+0%2C+-5%2C+0%2C+0%2C+0%2C+0%2C+-7%2C+-7%29+\"
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\n" ); document.write( "We now have our matrix in triangular form -- let's write the equations using the coefficients from it:
\n" ); document.write( "\"+x+%2B+y+%2B+z+=+2+\"
\n" ); document.write( "\"+-5y+=+0+\"
\n" ); document.write( "\"+-7z+=+-7+\"
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\n" ); document.write( "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:
\n" ); document.write( "\"+1+%2B+0+%2B+1+=+2+\"
\n" ); document.write( "\"+2%281%29+-+3%280%29+%2B+2%281%29+=+4+\"
\n" ); document.write( "\"+4%281%29+%2B+0+-+3%281%29+=+1+\"
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\n" ); document.write( "Hope this helps. ~ Joe \n" ); document.write( "
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