document.write( "Question 377111: Solve the system by elimination
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Algebra.Com's Answer #268061 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "The idea is to start with a system of 3 equations in 3 variables, then\r\n" );
document.write( "eliminate a variable and get a new system of only 2 equations in only 2\r\n" );
document.write( "variables.  Then we eliminate one of those and we end up with only 1 equation\r\n" );
document.write( "and 1 variable, (which is the first kind of equation we ever learned to solve\r\n" );
document.write( "when we just began studying algebra.)  \r\n" );
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document.write( "                      9x + 5y -  z = -11\r\n" );
document.write( "                      6x + 4y + 2z =   2\r\n" );
document.write( "                      2x - 2y + 4z =   4\r\n" );
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document.write( "Notice that it just so happens that the 2nd and 3rd equations can be divided\r\n" );
document.write( "through by 2, so we might as well do that from the start:\r\n" );
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document.write( "                      9x + 5y -  z = -11\r\n" );
document.write( "                      3x + 2y +  z =   1\r\n" );
document.write( "                       x -  y + 2z =   2\r\n" );
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document.write( "Pick a letter to eliminate and pick two of the equations that contain that\r\n" );
document.write( "letter to eliminate it from.  It is arbitrary which you pick, but you should\r\n" );
document.write( "pick them so as to minimize the amount of work.  I will pick the first two\r\n" );
document.write( "equations to eliminate z from:\r\n" );
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document.write( "We don't need to multiply either by anything because the terms in z will cancel\r\n" );
document.write( "if we add them -- corresponding term to corresponding term.\r\n" );
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document.write( "                      9x + 5y - z = -11\r\n" );
document.write( "                      3x + 2y + z =   1\r\n" );
document.write( "                     ------------------\r\n" );
document.write( "                     12x + 7y     = -10\r\n" );
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document.write( "Now we need another equation without a z-term.  So we pick the original\r\n" );
document.write( "equation that we haven't used and one of the ones we already used.  We have not\r\n" );
document.write( "used this equation, (the third original equation):\r\n" );
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document.write( "                       x -  y + 2z = 2\r\n" );
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document.write( "We will take it and the second original equation\r\n" );
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document.write( "                       x -  y + 2z =  2\r\n" );
document.write( "                      3x + 2y +  z =  1\r\n" );
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document.write( "(At this point BE SURE to eliminate the SAME variable z that was eliminated the\r\n" );
document.write( "first time).  We must eliminate z from these too, so we will multiply the\r\n" );
document.write( "bottom one by -2 to make the z-terms cancel when we add them term by term.\r\n" );
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document.write( "                       x -  y + 2z =  2\r\n" );
document.write( "                   -2(3x + 2y +  z =  1)\r\n" );
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document.write( "                       x -  y + 2z =  2\r\n" );
document.write( "                     -6x - 4y - 2z = -2\r\n" );
document.write( "                      -----------------\r\n" );
document.write( "                     -5x - 5y      =  0\r\n" );
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document.write( "This can be divided through by -5\r\n" );
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document.write( "                             x + y = 0 \r\n" );
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document.write( "Now we have a system of two equation and two variables:\r\n" );
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document.write( "                             x +  y =   0\r\n" );
document.write( "                           12x + 7y = -10\r\n" );
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document.write( "We can solve them any way we know how, but since you said elimination I will\r\n" );
document.write( "not switch over to substitution just yet.  Eliminate the y term by multiplying\r\n" );
document.write( "the first one through by -7 before adding term by term.\r\n" );
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document.write( "                          -7(x +  y =   0)\r\n" );
document.write( "                           12x + 7y = -10\r\n" );
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document.write( "                           -7x - 7y =   0\r\n" );
document.write( "                           12x + 7y = -10\r\n" );
document.write( "                           --------------\r\n" );
document.write( "                            5x      = -10\r\n" );
document.write( "                                  x = -2\r\n" );
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document.write( "Now we switch to substitution.  Substitute -2 for x in\r\n" );
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document.write( "                              x + y = 0\r\n" );
document.write( "                             -2 + y = 0\r\n" );
document.write( "                                  y = 2\r\n" );
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document.write( "Now substitute -2 for x and 2 for y in one of the original equations, say the\r\n" );
document.write( "second original equation:\r\n" );
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document.write( "                      3x + 2y + z = 1\r\n" );
document.write( "                 3(-2) + 2(2) + z = 1\r\n" );
document.write( "                       -6 + 4 + z = 1\r\n" );
document.write( "                           -2 + z = 1\r\n" );
document.write( "                                z = 3 \r\n" );
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document.write( "Solution:  (x,y,z) = (-2,2,3)\r\n" );
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document.write( "Edwin
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