document.write( "Question 1098559: x^2+xy+yz+zx=30
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Algebra.Com's Answer #713208 by ikleyn(52788) $\"\"$ $\"About$

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document.write( "Step 1.  Factor left side of each equation. You will get\r\n" );
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document.write( "    x^2+xy+yz+zx = 30    is the same as    (x+y)*(x+z) = 30    (1)\r\n" );
document.write( "    y^2+xy+yz+zx = 15    is the same as    (x+y)*(y+z) = 15    (2)\r\n" );
document.write( "    z^2+xy+yz+zx = 18    is the same as    (x+z)*(y+z) = 18    (3)\r\n" );
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document.write( "Step 2.  Multiply equations (1), (2) and (3)  (left sides and right sides). You will get\r\n" );
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document.write( "     = 30*15*18 = (15*2)*15*(2*3^2) =  =  = .\r\n" );
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document.write( "    Take the square root of both sides.  You will get  (x+y)*(x+z)*(y+z) = +/- ,   or\r\n" );
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document.write( "    (x+y)*(x+z)*(y+z) = +/- 90.    (4)\r\n" );
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document.write( "Step 3.  First, let us consider the case\r\n" );
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document.write( "    (x+y)*(x+z)*(y+z) = 90.        (5)\r\n" );
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document.write( "    Divide eq(5) by eq(1) (both sides).  You will get    y + z = 3.       (6)    (3 = 90/30)\r\n" );
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document.write( "    Divide eq(5) by eq(2) (both sides).  You will get    x + z = 6.       (7)    (6 = 90/15)\r\n" );
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document.write( "    Divide eq(5) by eq(3) (both sides).  You will get    x + y = 5.       (8)    (5 = 90/18)\r\n" );
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document.write( "Step 4.  To solve the system (6), (7), (8), add all three equation (6), (7) and (8) (both sides). You will get\r\n" );
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document.write( "    2x + 2y + 2z = 3 + 6 + 5 = 14.  It implies           x + y + z = 7.   (9)\r\n" );
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document.write( "    Now   subtract eq(6) from eq(9)  (both sides).   You will get  x = 7-3 = 4.\r\n" );
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document.write( "    Next, subtract eq(7) from eq(9)  (both sides).   You will get  y = 7-6 = 1.\r\n" );
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document.write( "    Next, subtract eq(7) from eq(9)  (both sides).   You will get  z = 7-5 = 2.\r\n" );
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document.write( "    So, (x,y,z) = (4,1,2) is the solution in this case.\r\n" );
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document.write( "Step 5.  Now let us consider the case\r\n" );
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document.write( "    (x+y)*(x+z)*(y+z) = -90.        \r\n" );
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document.write( "    Doing by the same way, you will obtain the solution  (x,y,z) = (-4,-1,-2)  in this case.\r\n" );
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document.write( "Step 6.  Check.  It is enough to check the positive solution only.\r\n" );
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document.write( "     \r\n" );
document.write( "    4^2 + 4*1 + 4*2 + 1*2 = 30.     ! Correct !\r\n" );
document.write( "    1^2 + 4*1 + 4*2 + 1*2 = 15.     ! Correct !\r\n" );
document.write( "    2^2 + 4*1 + 4*2 + 1*2 = 18.     ! Correct !\r\n" );
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document.write( "Answer.  The given system has two and only two solutions  (x,y,z) = (4,1,2)  and  (x,y,z) = (-4,-1,-2).\r\n" );
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document.write( "         Therefore,   =  = 21.\r\n" );
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