document.write( "Question 1149373: x − 2y + 3z + w = −6
\n" ); document.write( "x − 3y + z − w = −13
\n" ); document.write( "x − y = −4
\n" ); document.write( "x + z + w = 2
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Algebra.Com's Answer #770710 by Edwin McCravy(20063) $\"\"$ $\"About$
You can put this solution on YOUR website!
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document.write( "You didn't specify which of the many methods for solving systems of equations,\r\n" );
document.write( "so I just picked one at random.  Next time be sure to specify the method, for\r\n" );
document.write( "there are more than one way to solve lots of problems in algebra.  But\r\n" );
document.write( "regardless, the answer will be (x,y,z,w) = (1,5,1,0).\r\n" );
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document.write( "(1)   x − 2y + 3z + w =  −6\r\n" );
document.write( "(2)   x − 3y +  z − w = −13\r\n" );
document.write( "(3)   x −  y          =  −4\r\n" );
document.write( "(4)   x      +  z + w =   2\r\n" );
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document.write( "Since w is already missing from (3), let's eliminate it from\r\n" );
document.write( "(1) and (2) by adding them term by term:\r\n" );
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document.write( "(1)   x − 2y + 3z + w =  −6\r\n" );
document.write( "(2)   x − 3y +  z − w = −13\r\n" );
document.write( "---------------------------\r\n" );
document.write( "(5)  2x - 5y + 4z     = -19\r\n" );
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document.write( "Let's also eliminate w from (2) and (4) by adding them term by term:\r\n" );
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document.write( "(2)   x − 3y +  z − w = −13\r\n" );
document.write( "(4)   x      +  z + w =   2\r\n" );
document.write( "---------------------------\r\n" );
document.write( "(6)  2x - 3y + 2z     = -11\r\n" );
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document.write( "Now we have a new system of only 3 equations in 3 unknowns:\r\n" );
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document.write( "(3)   x −   y      =  −4\r\n" );
document.write( "(5)  2x -  5y + 4z = -19\r\n" );
document.write( "(6)  2x -  3y + 2z = -11 \r\n" );
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document.write( "Since z is also missing from (3), let's eliminate z from (5) and (6)\r\n" );
document.write( "by first multiplying (6) through by -2\r\n" );
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document.write( "Multiplying (6) through by -2 gives\r\n" );
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document.write( "(7) -4x +  6y - 4z =  22\r\n" );
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document.write( "Now we add (5) and (7) term by term:\r\n" );
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document.write( "(5)  2x -  5y + 4z = -19\r\n" );
document.write( "(7) -4x +  6y - 4z =  22\r\n" );
document.write( "------------------------\r\n" );
document.write( "(8) -2x +   y      =   3  \r\n" );
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document.write( "Now we have a new system of only 2 equations in 2 unknowns:\r\n" );
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document.write( "(3)   x −   y  =  −4\r\n" );
document.write( "(8) -2x +   y  =   3 \r\n" );
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document.write( "We can eliminate y just by adding (3) and (8)\r\n" );
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document.write( "(3)   x −   y  =  −4\r\n" );
document.write( "(8) -2x +   y  =   3 \r\n" );
document.write( "--------------------\r\n" );
document.write( "(9)  -x        =  -1\r\n" );
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document.write( "Now we have just one equation in one unknown, which we can solve\r\n" );
document.write( "by dividing both sides by -1\r\n" );
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document.write( "(10) x = 1\r\n" );
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document.write( "We substitute 1 for x in (3)\r\n" );
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document.write( "(3)   x −  y  =  −4\r\n" );
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document.write( "      1 −  y  =  −4      \r\n" );
document.write( "          -y  =  -5\r\n" );
document.write( "           y  =   5\r\n" );
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document.write( "We substitute 1 for x and 5 for y in (6) \r\n" );
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document.write( "(6)    2x - 3y   + 2z = -11  \r\n" );
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document.write( "(6)  2(1) - 3(5) + 2z = -11\r\n" );
document.write( "        2 -   15 + 2z = -11\r\n" );
document.write( "             -13 + 2z = -11\r\n" );
document.write( "                   2z = 2\r\n" );
document.write( "                    z = 1\r\n" );
document.write( "We substitute 1 for x and 1 for z in (4)\r\n" );
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document.write( "(4)   x  +  z  + w = 2\r\n" );
document.write( "     (1) + (1) + w = 2\r\n" );
document.write( "             2 + w = 2\r\n" );
document.write( "                 w = 0\r\n" );
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document.write( "So (x,y,z,w) = (1,5,1,0)\r\n" );
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document.write( "Edwin
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