document.write( "Question 984773: x^2=y+z, y^2=z+x, z^2=x+y, then what is the value of (1/(x+1)) + (1/(y+1)) + (1/(z+1))? \n" ); document.write( "
Algebra.Com's Answer #605609 by Edwin McCravy(20055) $\"\"$ $\"About$
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document.write( "x²=y+z,   y²=z+x,  z²=x+y\r\n" );
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document.write( "The three equations are symmetrical,\r\n" );
document.write( "so anything we discover about any pair\r\n" );
document.write( "of variables, it will also be true of any\r\n" );
document.write( "other pair of variables.\r\n" );
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document.write( "Solve the first two for z.\r\n" );
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document.write( "x²-y=z    y²-x=z   \r\n" );
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document.write( "x²-y = y²-x, since both equal to z\r\n" );
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document.write( "     x²-y² = -x+y\r\n" );
document.write( "(x-y)(x+y) = -(x-y)\r\n" );
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document.write( "We will see if we can divide by x-y\r\n" );
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document.write( "We will first assume x-y ≠ 0. Then we can divide both\r\n" );
document.write( "sides by x-y and get\r\n" );
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document.write( "      x+y = -1\r\n" );
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document.write( "But since z²=x+y, then that would \r\n" );
document.write( "mean z²=-1 and z would be ±i.\r\n" );
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document.write( "Then by the symmetry x and y would\r\n" );
document.write( "also be ±i  but then x²=y+z would give\r\n" );
document.write( "-1=i+i, -1=i-i, -1=-i+i or -1=-i-i,\r\n" );
document.write( "none of which are possible.  \r\n" );
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document.write( "Therefore we cannot divide both sides of\r\n" );
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document.write( "(x-y)(x+y) = -(x-y)\r\n" );
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document.write( "by x-y, and therefore the assumption x-y≠0 \r\n" );
document.write( "was false and therefore x-y=0 and x=y.\r\n" );
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document.write( "By symmetry x=y=z, so all three equations\r\n" );
document.write( "amount to\r\n" );
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document.write( "x²=x+x\r\n" );
document.write( "x²=2x\r\n" );
document.write( "x²-2x = 0\r\n" );
document.write( "x(x-2) = 0\r\n" );
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document.write( "x=0, x=2\r\n" );
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document.write( "By symmetry then x=y=z=0 or x=y=z=2\r\n" );
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document.write( "If x=y=z=0 then\r\n" );
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document.write( " =\r\n" );
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document.write( " =\r\n" );
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document.write( " =  = 3\r\n" );
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document.write( "If x=y=z=2 then\r\n" );
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document.write( " =\r\n" );
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document.write( " =\r\n" );
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document.write( " = 1\r\n" );
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document.write( "Answer: 3 or 1\r\n" );
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document.write( "Edwin
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