Are the positive integers x and y consecutive?
(1)\(x^2 - y^2 = 2y + 1\)
(2) \(x^2 - xy - x = 0\)
The question is basically asking whether \(x=y+1\)
\(x^2 - y^2=2y+1\) can be written as \((x+y)(x-y)=2y+1\).----equation 1
If we put \(x=y+1\), then LHS must be equal to RHS.
Equation 1 can be written, after substituting x=y+1, as \((2y+1)(1)=2y+1\). They are equal. Hence x and y are consecutive.
The above equation can be equal to 0 only when \(x-y=1\) because x is given to be positive.
Hope I am correct.
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