document.write( "Question 652720: My son and I are trying to figure out how to solve each system by elimination. When we try to put the matrice in the online matrice calculators, we can't figure out how to do it. We are also struggling to do this the long way. His first problem is:\r
\n" ); document.write( "\n" ); document.write( "Solve each system by elimination:
\n" ); document.write( "x+y+z=-1
\n" ); document.write( "2x-y+2z=-5
\n" ); document.write( "-x+2y-z=4\r
\n" ); document.write( "\n" ); document.write( "If you could explain how to do it the long way, and or how to put it in the online matrice calculator, would be a great help!
\n" ); document.write( "Thank you! \n" ); document.write( "
Algebra.Com's Answer #408106 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "  x +  y +  z = -1\r\n" );
document.write( " 2x -  y + 2z = -5\r\n" );
document.write( " -x + 2y -  z =  4\r\n" );
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document.write( "Add the first equation term-by-term to the third \r\n" );
document.write( "equation and the x's and z's will both cancel:\r\n" );
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document.write( "  x +  y +  z = -1\r\n" );
document.write( " -x + 2y -  z =  4\r\n" );
document.write( "------------------\r\n" );
document.write( "      3y      =  3\r\n" );
document.write( "       \r\n" );
document.write( "Divide both sides by 3 and you get\r\n" );
document.write( "            \r\n" );
document.write( "             y = 1\r\n" );
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document.write( "Substitute y = 1 in the first equation:\r\n" );
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document.write( "     x + y + z = -1\r\n" );
document.write( "     x + 1 + z = -1\r\n" );
document.write( "         x + z = -2\r\n" );
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document.write( "Substitute y = 1 in the second equation:\r\n" );
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document.write( "   2x - y + 2z = -5\r\n" );
document.write( "   2x - 1 + 2z = -5\r\n" );
document.write( "       2x + 2z = -4\r\n" );
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document.write( "Divide through by 2\r\n" );
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document.write( "        x + z = -2\r\n" );
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document.write( "Wow!  We got the same equation:\r\n" );
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document.write( "Substitute y = 1 in the third equation:\r\n" );
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document.write( "  -x + 2y - z =  4\r\n" );
document.write( "-x + 2(1) - z =  4\r\n" );
document.write( "   -x + 2 - z =  4\r\n" );
document.write( "       -x - z =  2\r\n" );
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document.write( "Divide through by -1\r\n" );
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document.write( "        x + z = -2\r\n" );
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document.write( "Wow!  We got the same equation a third time:  \r\n" );
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document.write( "This means the system is dependent and has infinitely\r\n" );
document.write( "many solutions.\r\n" );
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document.write( "The way we handle such an equation is to choose an\r\n" );
document.write( "arbitrary value k for the last letter z, that is z=k\r\n" );
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document.write( "We substitute k for z and have\r\n" );
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document.write( "       x + k = -2\r\n" );
document.write( "           x = -2 - k\r\n" );
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document.write( "Then we have the general solution\r\n" );
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document.write( "          x = -2-k, y=1, z=k\r\n" );
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document.write( " (x, y, z) = (-2-k, 1, k)  \r\n" );
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document.write( " There are many many solutions.  Here are some sample solutions:\r\n" );
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document.write( "If k = 0, we have the solution (x, y, z) = (-2, 1, 0)\r\n" );
document.write( "If k = 1, we have the solution (x, y, z) = (-3, 1, 1)\r\n" );
document.write( "If k = -2, we have the solution (x, y, z) = (0, 1, -2)\r\n" );
document.write( "If k = -7, we have the solution (x, y, z) = (5, 1, -7)\r\n" );
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document.write( "Edwin
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