document.write( "Question 1208609: Let x and y be integers. If x and y satisfy 37x + 82y = -172 and x + y = -1, then find x. \n" ); document.write( "

Algebra.Com's Answer #846975 by ikleyn(52787) $\"\"$ $\"About$

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\n" ); document.write( "Let x and y be integers. If x and y satisfy 37x + 82y = -172 and x + y = -1, then find x.
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document.write( "                    Solve by the Elimination method\r\n" );
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document.write( "Starting equations are\r\n" );
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document.write( "    37x + 82y  = -172,       (1)\r\n" );
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document.write( "      x +   y  =   -1.       (2)\r\n" );
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document.write( "They want you find x.  So, let's exclude y from the equations.\r\n" );
document.write( "For it, multiply equation (2) by 82. Keep equation (1) as is\r\n" );
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document.write( "    37x + 82y  = -172,       (1')\r\n" );
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document.write( "    82x + 82y =   -82.       (2')\r\n" );
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document.write( "Now subtract equation (1') from equation (2').  The terms with \"y\" will cancel each other, \r\n" );
document.write( "and you will get\r\n" );
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document.write( "    82x - 37x = -82 - (-172),\r\n" );
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document.write( "or\r\n" );
document.write( "         45x      =      90.\r\n" );
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document.write( "Hence, x = 90/45 = 2.\r\n" );
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document.write( "At this point, we formally completed the solution.\r\n" );
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document.write( "But we still need to check that the other solution, \"y\", is the integer number.\r\n" );
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document.write( "For it we substitute x= 2 into equation (2) and find\r\n" );
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document.write( "    y = -1 - x = -1 - 2 = -3.\r\n" );
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document.write( "Both \"x\" and \"y\" are integer, so, all the requirements are satisfied, and the  ANSWER  is  x= 2.\r\n" );
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document.write( "                    Solve by the Substitution method\r\n" );
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document.write( "From equation (2), express y = -1 - x and substitute it into equation (1).  You will get\r\n" );
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document.write( "    37x + 82*(-1-x)  = -172,\r\n" );
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document.write( "    37x - 82 - 82x     = -172,\r\n" );
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document.write( "     37x - 82x = -172 + 82\r\n" );
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document.write( "          -45x      =    -90\r\n" );
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document.write( "                x      =  = 2.\r\n" );
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document.write( "We get the same answer.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved in two different methods for your better understanding.\r
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