document.write( "Question 674385: Solve the system of equations in 3 variables. \r
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Algebra.Com's Answer #419241 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Line them up so the letters, signs and equal signs are in columns:\r\n" );
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document.write( "equation 1:     4x + 2y - 6z = -38\r\n" );
document.write( "equation 2:     5x - 4y +  z = -18\r\n" );
document.write( "equation 3:      x + 3y + 7z =  38\r\n" );
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document.write( "We choose the equation 1 and 2 to eliminate y from.\r\n" );
document.write( "If we multiply equation 1 by 2 the 2y will become 4y\r\n" );
document.write( "and cancel with the -4y in equation 2:\r\n" );
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document.write( "We multiply equation 1 by 2 and write equation 2 under it\r\n" );
document.write( "and add them term by term and call it equation 4:\r\n" );
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document.write( "                8x + 4y - 12z = -76\r\n" );
document.write( "equation 2:     5x - 4y +   z = -18\r\n" );
document.write( "-----------------------------------\r\n" );
document.write( "equation 4:    13x      - 11z = -94\r\n" );
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document.write( "Now we have to use the unused equation 3 with either of\r\n" );
document.write( "the equations we just used, 1 or 2, and eliminate that\r\n" );
document.write( "SAME variable, y.  I will pick equations 1 and 3\r\n" );
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document.write( "equation 1:     4x + 2y - 6z = -38\r\n" );
document.write( "equation 3:      x + 3y + 7z =  38\r\n" );
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document.write( "To eliminate y, we must make the y-terms cancel out.\r\n" );
document.write( "Their coefficients are 2 and 3.  The smallest whole\r\n" );
document.write( "number 2 and 3 will both go into is 6.  So we want\r\n" );
document.write( "to make the 2y term into +6y and the 3y term into -6y.\r\n" );
document.write( "So we multiply equation 1 by 3 and equation 2 by -2, and\r\n" );
document.write( "add them term by term, and call it equation 5:\r\n" );
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document.write( "               12x + 6y - 18z = -114\r\n" );
document.write( "               -2x - 6y - 14z =  -76\r\n" );
document.write( "              ----------------------\r\n" );
document.write( "equation 5     10x      - 32z = -190\r\n" );
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document.write( "Now we take equations 4 and 5:\r\n" );
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document.write( "equation 4:    13x - 11z =  -94\r\n" );
document.write( "equation 5     10x - 32z = -190\r\n" );
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document.write( "Notice that equation 5 can be divided through by 2.\r\n" );
document.write( "That will make it a little easier. We'll call it equation 6.\r\n" );
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document.write( "equation 4:    13x - 11z = -94\r\n" );
document.write( "equation 6:     5x - 16z = -95\r\n" );
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document.write( "We'll pick one of the letters x or y to eliminate.\r\n" );
document.write( "Let's pick x to eliminate. To eliminate x, we must \r\n" );
document.write( "make the x-terms cancel out. Their coefficients are \r\n" );
document.write( "13 and 5.  The smallest whole number 13 and 5 will \r\n" );
document.write( "both go into is 65.  So we want to make the 13x term \r\n" );
document.write( "into +65x and the 5x term into -65x.\r\n" );
document.write( "So we multiply equation 4 by 5 and equation 6 by -13, and\r\n" );
document.write( "add them term by term, and solve for y:\r\n" );
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document.write( "               65x -  55z = -470\r\n" );
document.write( "              -65x + 208z = 1235\r\n" );
document.write( "              ------------------\r\n" );
document.write( "                     153z = 765\r\n" );
document.write( "                        z = \"765%2F153\"\r\n" );
document.write( "                        z = 5\r\n" );
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document.write( "Now we substitute z = 5 into equation 5 or 6.  We'll\r\n" );
document.write( "choose equation 6:\r\n" );
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document.write( "equation 6:     5x - 16z = -95\r\n" );
document.write( "              5x - 16(5) = -95\r\n" );
document.write( "                 5x - 80 = -95\r\n" );
document.write( "                      5x = -15\r\n" );
document.write( "                       x = \"%28-15%29%2F5\"\r\n" );
document.write( "                       x = -3\r\n" );
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document.write( "Finally we substitute x = -3 and z = 5 into any one\r\n" );
document.write( "of the original equations 1,2, or 3.  We'll choose\r\n" );
document.write( "equation 1:\r\n" );
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document.write( "equation 1:     4x + 2y - 6z = -38\r\n" );
document.write( "           4(-3) + 2y - 6(5) = -38\r\n" );
document.write( "               -12 + 2y - 30 = -38\r\n" );
document.write( "                    -42 + 2y = -38\r\n" );
document.write( "                          2y = 4\r\n" );
document.write( "                           y = \"4%2F2\"\r\n" );
document.write( "                           y = 2\r\n" );
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document.write( "So the solution is (x,y,z) = (-3,2,5)\r\n" );
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document.write( "Edwin
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