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document.write( "These two equalities \r\n" );
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document.write( "x + y + z = 7 (1) and\r\n" );
document.write( "xy + yz + zx = 11 (2)\r\n" );
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document.write( "imply\r\n" );
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= 
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= ,\r\n" );
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document.write( "
+ 2*11 = 49,\r\n" );
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document.write( " = 49 - 22 = 27.\r\n" );
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document.write( "I am repeating it again: (1) and (2) imply\r\n" );
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document.write( " = 27. (3)\r\n" );
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document.write( "Equation (3) defines the sphere in 3D space.\r\n" );
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document.write( "Equation (1) represents the plane in 3D.\r\n" );
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document.write( " Therefore, I can state that equations (1) and (2) define the section of the sphere (3) by the plane (1).\r\n" );
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document.write( " Having this geometric interpretation, we can turn ON our geometric intuition.\r\n" );
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document.write( "It becomes clear that the maximum and the minimum of \"z\" are achieved at the plane x = y.\r\n" );
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document.write( "The section of the sphere = 27 by the plane x = y is the circle of the radius of 
centered at the origin \r\n" );
document.write( "of the coordinate system (every section of a sphere by a plane through it center is the great circle of the sphere).\r\n" );
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document.write( " Therefore, the maximal value of z is the intersection point of this circle section at the plane x = y and the straight line L\r\n" );
document.write( " which is the intersection of the plane x+y+z = 7 and the plane x = y. (Obviously, the same is true for the minimal value of z, too).\r\n" );
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document.write( "Let us go to this plane x = y and introduce the axis and the coordinate \"u\" in this plane orthogonal to z-axis.\r\n" );
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document.write( "In this plane the equation of the circle is\r\n" );
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= 27 (4)\r\n" );
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document.write( "and the equation of the straight line L is\r\n" );
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document.write( "z = 
(5)\r\n" );
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document.write( "Thus to find 
we need to solve the system of two equations (4), (5).\r\n" );
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document.write( "So make the substitution, simplify . . . and you will get . . . (I just did everything for you . . . )\r\n" );
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document.write( "
= 
.\r\n" );
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document.write( "The smaller root for u is 
= 
, and the corresponding value of is = 
= 7 - 2 = 5.\r\n" );
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document.write( "The larger root for u is 
= 
, and the corresponding value of 
is = 
= 
= 
= 
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document.write( "Answer. = 5. = .\r
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document.write( "Solved.\r
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