document.write( "Question 164381: system of equation with four variables.
\n" ); document.write( "x + y + z + w = 0
\n" ); document.write( "x + y - z - w = 1
\n" ); document.write( "x - y + z + w = 3
\n" ); document.write( "x - y - z + w = -1 \n" ); document.write( "
Algebra.Com's Answer #121110 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "system of equation with four variables.\r\n" );
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document.write( "x + y + z + w =  0\r\n" );
document.write( "x + y - z - w =  1\r\n" );
document.write( "x - y + z + w =  3\r\n" );
document.write( "x - y - z + w = -1\r\n" );
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document.write( "We can eliminate y, z and w simply\r\n" );
document.write( "by adding the second and third equations,\r\n" );
document.write( "term by term\r\n" );
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document.write( " x + y - z - w =  1\r\n" );
document.write( " x - y + z + w =  3\r\n" );
document.write( "-------------------\r\n" );
document.write( "2x             =  4\r\n" );
document.write( "            2x =  4 \r\n" );
document.write( "             x =  2\r\n" );
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document.write( "Substitute 2 for x in the\r\n" );
document.write( "original equations.\r\n" );
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document.write( "2 + y + z + w =  0\r\n" );
document.write( "2 + y - z - w =  1\r\n" );
document.write( "2 - y + z + w =  3\r\n" );
document.write( "2 - y - z + w = -1\r\n" );
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document.write( " y + z + w = -2\r\n" );
document.write( " y - z - w = -1\r\n" );
document.write( "-y + z + w =  1\r\n" );
document.write( "-y - z + w = -3\r\n" );
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document.write( "The third equation is just the\r\n" );
document.write( "second equation multiplied by\r\n" );
document.write( "-1, so we can eliminate it:\r\n" );
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document.write( " y + z + w = -2\r\n" );
document.write( " y - z - w = -1\r\n" );
document.write( "-y - z + w = -3\r\n" );
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document.write( "We can eliminate both z and w\r\n" );
document.write( "by adding the first and second\r\n" );
document.write( "equation:\r\n" );
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document.write( " y + z + w = -2\r\n" );
document.write( " y - z - w = -1\r\n" );
document.write( "---------------\r\n" );
document.write( "2y         = -3\r\n" );
document.write( "        2y = -3\r\n" );
document.write( "         y = \"-3%2F2\"  \r\n" );
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document.write( "Substitute \"-3%2F2\" for y\r\n" );
document.write( "in the three: \r\n" );
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document.write( " \"-3%2F2\" + z + w = -2\r\n" );
document.write( " \"-3%2F2\" - z - w = -1\r\n" );
document.write( "\"-%28-3%2F2%29\" - z + w = -3\r\n" );
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document.write( " \"-3%2F2\" + z + w = -2\r\n" );
document.write( " \"-3%2F2\" - z - w = -1\r\n" );
document.write( "  \"3%2F2\" - z + w = -3\r\n" );
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document.write( "Multiply each through by 2 to\r\n" );
document.write( "eliminate the fractions:\r\n" );
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document.write( "  -3 + 2z + 2w = -4\r\n" );
document.write( "  -3 - 2z - 2w = -2\r\n" );
document.write( "   3 - 2z + 2w = -6\r\n" );
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document.write( "       2z + 2w = -1 \r\n" );
document.write( "      -2z - 2w =  1\r\n" );
document.write( "      -2z + 2w = -9\r\n" );
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document.write( "The second equation is just the\r\n" );
document.write( "first equation multiplied by -1,\r\n" );
document.write( "so we can eliminate it:\r\n" );
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document.write( "       2z + 2w = -1 \r\n" );
document.write( "      -2z + 2w = -9\r\n" );
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document.write( "Now we can eliminate z by adding\r\n" );
document.write( "those two equations:\r\n" );
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document.write( "       2z + 2w =  -1 \r\n" );
document.write( "      -2z + 2w =  -9\r\n" );
document.write( "     ---------------\r\n" );
document.write( "            4w = -10\r\n" );
document.write( "             w = \"-10%2F4\"\r\n" );
document.write( "             w = \"-5%2F2\"\r\n" );
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document.write( "So (x,y,z,w) = (2,\"-3%2F2\",2,\"-5%2F2\")\r\n" );
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document.write( "Edwin
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