document.write( "Question 574555: What are all ordered triples of numbers (x, y, z) which satisfy the system of equations\r
\n" ); document.write( "\n" ); document.write( "(x + y)(z + 1) = 24,
\n" ); document.write( "(x + y)(y + 1) = 24, and
\n" ); document.write( "(y + z)(x + 1) = 24? \n" ); document.write( "
Algebra.Com's Answer #369197 by KMST(5328)\"\" \"About 
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\"%28x+%2B+y%29%28z+%2B+1%29+=+24\"
\n" ); document.write( "\"%28x+%2B+y%29%28y+%2B+1%29+=+24\", and
\n" ); document.write( "\"%28y+%2B+z%29%28x+%2B+1%29+=+24\"
\n" ); document.write( "Combining the first two equations
\n" ); document.write( "\"%28x+%2B+y%29%28z+%2B+1%29+=+%28x+%2B+y%29%28y+%2B+1%29=24\"
\n" ); document.write( "So \"x%2By\" cannot be zero and \"z=y\"
\n" ); document.write( "If \"z=y\", the third equation turns into
\n" ); document.write( "\"%28y+%2B+y%29%28x+%2B+1%29+=+24\" ---> \"2y%28x%2B1%29=24\" --> \"y%28x%2B1%29=12\" --> \"yx%2By%29=12\" which can be substituted into one of the other equations
\n" ); document.write( "\"%28x+%2B+y%29%28y+%2B+1%29+=+24\" --> \"xy%2Bx%2By%5E2%2By=24\" --> \"%28xy%2By%29%2By%5E2%2Bx=24\" --> \"12%2By%5E2%2Bx=24\" --> \"y%5E2%2Bx=12\" --> \"x=12-y%5E2\"
\n" ); document.write( "Substituting \"x=12-y%5E2\" into \"y%28x%2B1%29=12\" we get
\n" ); document.write( "\"y%2813-y%5E2%29=12\" --> \"13y-y%5E3=12\" --> \"y%5E3-13y%2B12=0\"
\n" ); document.write( "That can be solve by factoring, starting by factoring out \"y-1\", because it's obvious that \"y=1\" is a solution
\n" ); document.write( "\"y%5E3-13y%2B12=0\" --> \"%28y-1%29%28y%5E2%2By-12%29=0\" --> \"%28y-1%29%28y%2B4%29%28y-3%29=0\"
\n" ); document.write( "SO the solutions are
\n" ); document.write( "\"z=y=1\" --> \"x=12-1%5E2=11\"
\n" ); document.write( "\"z=y=3\" --> \"x=12-3%5E2=12-9=3\"
\n" ); document.write( "\"z=y=-4\" --> \"x=12-%28-4%29%5E2=12-16=-4\"
\n" ); document.write( "The solutions would be (11,1,1), (3,3,3) and (-4,-4,-4).
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