document.write( "Question 549659: Hello,
\n" ); document.write( "I have the following system of equations. When I first tried to solve the problem using elimination (getting rid of y in 1&2), the whole problem canceled out. The same thing happened when I tried to eliminate z. I did finally solve the problem but only after trial and error using elimination.
\n" ); document.write( "Is there a way to recognize which variables are best to eliminate or at least which ones can not be eliminated?
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\n" ); document.write( "\n" ); document.write( "2x-2y-4z=-2
\n" ); document.write( "3x-3y-6z=-3
\n" ); document.write( "-2x+3y+z=7 \n" ); document.write( "

Algebra.Com's Answer #357946 by Edwin McCravy(20056) $\"\"$ $\"About$

You can put this solution on YOUR website!

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document.write( " 2x - 2y - 4z = -2\r\n" );
document.write( " 3x - 3y - 6z = -3\r\n" );
document.write( "-2x + 3y +  z  = 7\r\n" );
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document.write( "It is true that if you try to eliminate any variable in 1&2\r\n" );
document.write( "everything cancels out.  This means there are infinitely \r\n" );
document.write( "many solutions, but we can still find the general solution.\r\n" );
document.write( "So try to eliminate a letter using 2&3.  Notice that the y's\r\n" );
document.write( "will cancel if you add 2&3 as they are:\r\n" );
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document.write( " 3x - 3y - 6z = -3\r\n" );
document.write( "-2x + 3y +  z  = 7\r\n" );
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document.write( "  x      - 5z  = 4\r\n" );
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document.write( "So solve for  x:\r\n" );
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document.write( "             x = 4 + 5z\r\n" );
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document.write( "Substitute 4 + 5z for x in either 1 or 2. I'll pick 1\r\n" );
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document.write( "     2x - 2y - 4z = -2\r\n" );
document.write( "2(4+5z) - 2y - 4z = -2 \r\n" );
document.write( "8 + 10z - 2y - 4z = -2\r\n" );
document.write( "      8 + 6z - 2y = -2\r\n" );
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document.write( "Solve for y:\r\n" );
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document.write( "              -2y = -10 - 6z\r\n" );
document.write( "                y =  - \r\n" );
document.write( "                y = 5 + 3z\r\n" );
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document.write( "So the general solution is\r\n" );
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document.write( "x = 4 + 5z, y = 5 + 3z,  z = z\r\n" );
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document.write( "often written as a 3D point:\r\n" );
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document.write( "(x, y, z) = (4+5z, 5+3z, z)\r\n" );
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document.write( "There are infinitely many solutions.\r\n" );
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document.write( "To get some of them, substitute arbitrary values for z:\r\n" );
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document.write( "Substitute, say, 1 for z:\r\n" );
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document.write( "(x, y, z) = (4+5z, 5+3z, z) = (4+5·1, 5+3·1, z) = (9, 8, 1)\r\n" );
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document.write( "Substitute, say, -2 for z:\r\n" );
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document.write( "(x, y, z) = (4+5z, 5+3z, z) = (4+5·-2, 5+3·-2, z) = (4-10, 5-6, -2) = (-6, -1, -2)\r\n" );
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document.write( "You can get as many different solutions as you like just by \r\n" );
document.write( "substituting arbitrary values for z.\r\n" );
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document.write( "Edwin

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