document.write( "Question 651023: how would you solve these systems\r
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Algebra.Com's Answer #407545 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Line them up so that like letters line up vertically, you might\r\n" );
document.write( "like to put 1 coefficients before letters not showing a coefficient:\r\n" );
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document.write( "eq.1    2x-1y+1z = 3\r\n" );
document.write( "eq.2    3x-3y-1z = 4\r\n" );
document.write( "eq.3    1x+3y+3z = 6\r\n" );
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document.write( "I.  Pick a pair of equations and a letter that both contain and \r\n" );
document.write( "    eliminate that letter from them.\r\n" );
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document.write( "I pick eq.1 and eq.2 and z to eliminate from them:\r\n" );
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document.write( "eq.1    2x-1y+1z = 3\r\n" );
document.write( "eq.2    3x-3y-1z = 4\r\n" );
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document.write( "The coefficients are z in eq.1 and eq.2 are 1 and -1 which will \r\n" );
document.write( "cancel if we add corresponding terms in the two equations:\r\n" );
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document.write( "eq.1    2x-1y+1z = 3\r\n" );
document.write( "eq.2    3x-3y-1z = 4 \r\n" );
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document.write( "eq.4    5x-4y    = 7\r\n" );
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document.write( "II. Use the equation you haven't used yet with one of the other\r\n" );
document.write( "    equations and eliminate that SAME letter from them.\r\n" );
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document.write( "We haven't used eq.3 yet, so we put eq.3 with, say eq.2, and we\r\n" );
document.write( "eliminate the same variable that we eliminated in the first step: \r\n" );
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document.write( "eq.2    3x-3y-1z = 4\r\n" );
document.write( "eq.3    1x+3y+3z = 6\r\n" );
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document.write( "[Caution: Even though the y's would cancel if we added them just\r\n" );
document.write( "as they are, DO NOT add them, because we must eliminate the SAME \r\n" );
document.write( "letter we eliminated before, which is z, not y.]\r\n" );
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document.write( "To eliminate z we must multiply both sides of eq.2 by 3 to make\r\n" );
document.write( "the -1z term, become -3z, and then add them\r\n" );
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document.write( "        9x-9y-3z = 12\r\n" );
document.write( "eq.3    1x+3y+3z =  6\r\n" );
document.write( "        -------------\r\n" );
document.write( "       10x-6y    = 18\r\n" );
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document.write( "That can be divided through by 2, so we will do that:\r\n" );
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document.write( "eq.5   5x-3y = 9\r\n" );
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document.write( "III.  Solve the resulting 2×2 system of equations:\r\n" );
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document.write( "So we put eq.5 with eq.4:\r\n" );
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document.write( "eq.4   5x-4y = 7\r\n" );
document.write( "eq.5   5x-3y = 9\r\n" );
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document.write( "The 5x in eq.4 can be made to cancel the 5x in eq.5 by\r\n" );
document.write( "multiplying eq.4 by -1\r\n" );
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document.write( "      -5x+4y = -7\r\n" );
document.write( "eq.5   5x-3y =  9\r\n" );
document.write( "      -----------\r\n" );
document.write( "           y =  2\r\n" );
document.write( "        \r\n" );
document.write( "Now we substitute in either eq.4 or eq.5 to find x. Say,\r\n" );
document.write( "pick eq.5 and substitute y = 2\r\n" );
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document.write( "eq.5    5x-3y = 9\r\n" );
document.write( "      5x-3(2) = 9\r\n" );
document.write( "         5x-6 = 9\r\n" );
document.write( "           5x = 15\r\n" );
document.write( "            x = 3\r\n" );
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document.write( "IV.  Substitute the values obtained into one of the original equations\r\n" );
document.write( "     to find the remaining variable, which was the firet ofne eliminated.\r\n" );
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document.write( "So we substitute x = 4 and y = 2 in any of the original 3 equations \r\n" );
document.write( "to find z, which was the first letter we eliminated.  I'll use eq.1:\r\n" );
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document.write( "eq.1    2x-1y+1z = 3\r\n" );
document.write( "    2(3)-1(2)+1z = 3\r\n" );
document.write( "           6-2+z = 3\r\n" );
document.write( "             4+z = 3\r\n" );
document.write( "               z = -1\r\n" );
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document.write( "Solution (x,y,z) = (4,2,-1)\r\n" );
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document.write( "Edwin
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