document.write( "Question 1087439: Solving linear systems with 3 variables
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Algebra.Com's Answer #701770 by Edwin McCravy(20056) $\"\"$ $\"About$
You can put this solution on YOUR website!
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document.write( "He did it my Cramer's rule (determinants). I didn't \r\n" );
document.write( "know if you'd had that or just the elimination method.\r\n" );
document.write( "So I thought I'd do it by the first method that's\r\n" );
document.write( "usually taught.\r\n" );
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document.write( "Line them up so that like letters, signs, \r\n" );
document.write( "and equal signs, and numbers line up\r\n" );
document.write( "vertically.\r\n" );
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document.write( "4x - 3y      = 1\r\n" );
document.write( "     2y - 3z = 2\r\n" );
document.write( "3x      + 2z = 3\r\n" );
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document.write( "The idea is to reduce that to a linear system with\r\n" );
document.write( "only 2 variables.\r\n" );
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document.write( "Notice that one of the variables has already been \r\n" );
document.write( "eliminated in all three.  Pick one of the three\r\n" );
document.write( "to be one of the equations in the system of 2\r\n" );
document.write( "equations with 2 variables.  It doesn't matter \r\n" );
document.write( "which of the three you pick.  Say we pick the \r\n" );
document.write( "third one, from which y has already been \r\n" );
document.write( "eliminated.\r\n" );
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document.write( " 3x + 2z = 3\r\n" );
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document.write( "Now even though a variable has been eliminated from\r\n" );
document.write( "the other two, you still must eliminate the same\r\n" );
document.write( "variable from the other two that is already eliminated \r\n" );
document.write( "in the one we picked, 3x + 2z = 3.  So we eliminate\r\n" );
document.write( "y from the other two equations:\r\n" );
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document.write( "4x - 3y      = 1\r\n" );
document.write( "     2y - 3z = 2\r\n" );
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document.write( "To eliminate y from them, we multiply the first\r\n" );
document.write( "equation by 2 and the second one by 3, and then\r\n" );
document.write( "add them:\r\n" );
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document.write( "8x - 6y      = 2\r\n" );
document.write( "     6y - 9z = 6\r\n" );
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document.write( "8x      - 9z = 8 \r\n" );
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document.write( "So the linear system in 2 variables is\r\n" );
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document.write( "3x + 2z = 3\r\n" );
document.write( "8x - 9z = 8\r\n" );
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document.write( "We can eliminate z by multiplying the first one just\r\n" );
document.write( "above by 9 and the second one just above by 2\r\n" );
document.write( "and adding:\r\n" );
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document.write( "27x + 18z = 27\r\n" );
document.write( "16x - 18z = 16\r\n" );
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document.write( "43x       = 43\r\n" );
document.write( "        x = 1\r\n" );
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document.write( "Substitute x = 1 in\r\n" );
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document.write( "  3x + 2z = 3\r\n" );
document.write( "3(1) + 2z = 3\r\n" );
document.write( "   3 + 2z = 3\r\n" );
document.write( "       2z = 0\r\n" );
document.write( "        z = 0\r\n" );
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document.write( "Substitute x = 1 in the first original \r\n" );
document.write( "equation:\r\n" );
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document.write( "  4x - 3y = 1\r\n" );
document.write( "4(1) - 3y = 1\r\n" );
document.write( "   4 - 3y = 1\r\n" );
document.write( "      -3y = -3\r\n" );
document.write( "        y = 1 \r\n" );
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document.write( "Answer (x,y,z) = (1,1,0)\r\n" );
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document.write( "Edwin
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