document.write( "Question 450089:
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\n" ); document.write( "\n" ); document.write( " y + z = 7\r
\n" ); document.write( "\n" ); document.write( " x – z = 2 \n" ); document.write( "
Algebra.Com's Answer #309658 by Edwin McCravy(20063)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Solve   x + y = 9\r\n" );
document.write( "        y + z = 7\r\n" );
document.write( "        x – z = 2\r\n" );
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document.write( "That is a dependent system and has an infinite\r\n" );
document.write( "number of solutions.  I'll show you why that is\r\n" );
document.write( "true, but we start out b y the elimination method:\r\n" );
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document.write( "Begin by writing the equations neatly so that \r\n" );
document.write( "like terms and equal signs line up vertically:\r\n" );
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document.write( "x + y     = 9\r\n" );
document.write( "    y + z = 7\r\n" );
document.write( "x     – z = 2\r\n" );
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document.write( "The idea is to create from this system of\r\n" );
document.write( "3 equations in the same 3 unknowns to a system \r\n" );
document.write( "of two equations in the same two unknowns down \r\n" );
document.write( "to just 1 equation in 1 unknown, which we can\r\n" );
document.write( "solve for a number and substitute it back in\r\n" );
document.write( "previous equations to get the other two\r\n" );
document.write( "unknowns.\r\n" );
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document.write( "The first equation has only 2 unknowns x and\r\n" );
document.write( "y.  We can get another equations in only those\r\n" );
document.write( "same two unknowns by adding the 2nd and 3rd\r\n" );
document.write( "equations, so the the z's will cancel out:\r\n" );
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document.write( "    y + z = 7\r\n" );
document.write( "x     – z = 2\r\n" );
document.write( "-------------\r\n" );
document.write( "x + y     = 9\r\n" );
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document.write( "So now we have two equations in the same two\r\n" );
document.write( "variables and no others.\r\n" );
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document.write( "x + y     = 9\r\n" );
document.write( "x + y     = 9\r\n" );
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document.write( "But, strangely enough, they are the same equation, \r\n" );
document.write( "and so the original system is dependent and there are \r\n" );
document.write( "infinitely many solutions.\r\n" );
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document.write( "Solve that for y\r\n" );
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document.write( "y = 9 - x\r\n" );
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document.write( "and solve  y + z = 7 for z\r\n" );
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document.write( "z = 7 - y\r\n" );
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document.write( "Now to find one solution:\r\n" );
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document.write( "For instance pick x = 1, and substitute in y = 9 - x\r\n" );
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document.write( "y = 9 - 1\r\n" );
document.write( "y = 8\r\n" );
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document.write( "Substitute that in z = 7 - y\r\n" );
document.write( "  \r\n" );
document.write( "z = 7 - 8\r\n" );
document.write( "z = -1\r\n" );
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document.write( "So one of many solutions is (x,y,z) = (1,8,-1)\r\n" );
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document.write( "For another solution, I'll arbitrarily pick x = -3, and \r\n" );
document.write( "substitute that in y = 9 - x\r\n" );
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document.write( "y = 9 - (-3)\r\n" );
document.write( "y = ( + 3\r\n" );
document.write( "y = 12\r\n" );
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document.write( "Substitute that in z = 7 - y\r\n" );
document.write( "  \r\n" );
document.write( "z = 7 - 12\r\n" );
document.write( "z = -5\r\n" );
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document.write( "So another of the many possible solutions is (x,y,z) = (-3,12,-5)\r\n" );
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document.write( "Choose any values for x you like and the use those two equations to \r\n" );
document.write( "find y and z and find as many solutions to this system as you like.\r\n" );
document.write( "They'll always check in the original equations.\r\n" );
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document.write( "Edwin
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