Mo2men wrote:1- Does this mean that the DERIVED conclusion equation is wrong?
The derived equation is valid in the following sense:
If x and y are values such that
their product is an integer and
their sum is an integer -- as required by the two statements -- then we can conclude that x²+y² = integer.
x=y=√2 is irrelevant because it violates the condition in red.
The algebra performed in your solution is correct, but it seems time-consuming and not especially helpful.
Even if we correctly deduce that x²+y² = integer, we may still consider only cases that satisfy the two colored conditions above.
It seems more efficient to try to identify cases that satisfy these conditions directly, as in my solution above.
2- What is your advice when combined statements and choose plug-in values?
If your first case yields an answer of YES to the question stem, ask yourself how a second case could yield an answer of NO.
Here, an answer of YES is yielded if x and y are integers.
Subsequent cases should be such that x and y are NOT integers (fractions or roots).
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