document.write( "Question 418129: How do you solve this problem?
\n" ); document.write( "x+y+z-w=2
\n" ); document.write( "3x+y-z+w=8
\n" ); document.write( "x-5y+2z+w=1
\n" ); document.write( "-x-y+z+3w=4 \n" ); document.write( "
Algebra.Com's Answer #292732 by Edwin McCravy(20056) $\"\"$ $\"About$
You can put this solution on YOUR website!
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document.write( "There are many ways to solve it.  You didn't specify which way.\r\n" );
document.write( "I arbitrarily picked elimination.  Which method are you studying?\r\n" );
document.write( "Gaussian elimination?  Cramer's rule? If you're studying another \r\n" );
document.write( "method, then post again telling what method.  We aren't mind\r\n" );
document.write( "readers.\r\n" );
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document.write( " x +  y +  z -  w = 2\r\n" );
document.write( "3x +  y -  z +  w = 8\r\n" );
document.write( " x - 5y + 2z +  w = 1\r\n" );
document.write( "-x -  y +  z + 3w = 4\r\n" );
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document.write( "1. Pick a letter to eliminate.  I'll pick w\r\n" );
document.write( "2. Pick two equations to eliminate it from.  I pick the eq. 1 and eq. 2 \r\n" );
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document.write( " x +  y +  z -  w =  2\r\n" );
document.write( "3x +  y -  z +  w =  8\r\n" );
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document.write( "4x + 2y           = 10\r\n" );
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document.write( "It can be divided through by 2\r\n" );
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document.write( "           2x + y = 5\r\n" );
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document.write( "3. Pick two more equations to eliminate z from.  \r\n" );
document.write( "   I pick the eq. 2 and eq. 4\r\n" );
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document.write( "3x +  y -  z +  w =  8\r\n" );
document.write( "-x -  y +  z + 3w =  4\r\n" );
document.write( "----------------------\r\n" );
document.write( "2x           + 4w = 12\r\n" );
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document.write( "It can also be divided through by 2\r\n" );
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document.write( "           x + 2w = 6\r\n" );
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document.write( "3. Pick two more equations to eliminate z from. \r\n" );
document.write( "   But we must be sure this time to pick the equation we \r\n" );
document.write( "   haven't picked before. \r\n" );
document.write( "   I pick the eq. 2 and eq. 3\r\n" );
document.write( "  \r\n" );
document.write( "3x +  y -  z +  w = 8\r\n" );
document.write( " x - 5y + 2z +  w = 1\r\n" );
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document.write( "We must multiply the top one by 2 to make the z's cancel:\r\n" );
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document.write( "6x + 2y - 2z + 2w = 16\r\n" );
document.write( " x - 5y + 2z +  w =  1\r\n" );
document.write( "----------------------\r\n" );
document.write( "7x - 3y      + 3w = 17\r\n" );
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document.write( "Now we have this system\r\n" );
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document.write( "2x +  y      =  5\r\n" );
document.write( " x      + 2w =  6\r\n" );
document.write( "7x - 3y + 3w = 17\r\n" );
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document.write( "Since y is already eliminated from the 2nd eq., we will\r\n" );
document.write( "eliminate y from the 1st and 3rd.\r\n" );
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document.write( "2x +  y      =  5\r\n" );
document.write( "7x - 3y + 3w = 17\r\n" );
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document.write( "We must multiply the top one by 3 to make the y's cancel:\r\n" );
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document.write( " 6x + 3y      = 15\r\n" );
document.write( " 7x - 3y + 3w = 17\r\n" );
document.write( "------------------\r\n" );
document.write( "13x      + 3w = 32\r\n" );
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document.write( "Now we put that with the 2nd eq. and we have this system:\r\n" );
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document.write( "       x + 2w =  6\r\n" );
document.write( "     13x + 3w = 32\r\n" );
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document.write( "We must multiply the top one by -13 to make the x's cancel:\r\n" );
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document.write( "    -13x - 26w = -78\r\n" );
document.write( "     13x +  3w =  32\r\n" );
document.write( "   -----------------\r\n" );
document.write( "          -23w = -46\r\n" );
document.write( "             w =  2\r\n" );
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document.write( "Substitute w = 2 in\r\n" );
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document.write( "       x + 2w = 6\r\n" );
document.write( "     x + 2(2) = 6\r\n" );
document.write( "        x + 4 = 6\r\n" );
document.write( "            x = 2\r\n" );
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document.write( "Substitute x = 2 in\r\n" );
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document.write( "       2x + y = 5\r\n" );
document.write( "     2(2) + y = 5 \r\n" );
document.write( "        4 + y = 5\r\n" );
document.write( "            y = 1\r\n" );
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document.write( "Substitute x = 2, y = 1, and w = 2 in\r\n" );
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document.write( "x + y + z - w = 2\r\n" );
document.write( "2 + 1 + z - 2 = 2\r\n" );
document.write( "        1 + z = 2\r\n" );
document.write( "            z = 1\r\n" );
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document.write( "Solution: (x,y,z,w) = (2,1,1,2)\r\n" );
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document.write( "Edwin
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