Integer Solutions
Let , and be the integer solutions to the two simultaneous equations What is the value of
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1 solution
x^2 - 2xy + 4yz - 4z^2 = (x - 2z) * (x - 2y + 2z) = 1.
Thus either x - 2z = 1 = x - 2y + 2z or x - 2z = -1 = x - 2y + 2z.
For the first option we have that 2x - 2y = 2 -----> x - y = 1.
Thus y = x - 1 and z = (1/2) * ( x - 1). Plug this into x + y + z = 21 to get
x + (x - 1) + (1/2) * ( x - 1) = (5/2) * x - (3/2) = 21 ------->
5x - 3 = 42 ------> 5x = 45 ------> x = 9, which gives us y = 8 and z = 4.
Therefore x + 2y + 3z = 9 + 16 + 12 = 37.
The second option yields non-integer solutions.