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I have the following expression $GCD[d b(x-y)+a(d-x y), a(x-y)+b(d-x y)]$ where x and y are variables and a and b are co-prime. Now, if I eliminate the variable $x$ from the first expression of the GCD i.e. $d b(x-y)+a(d-x y)$ , I have $GCD[(y^2-d)*(a^2-d b^2), a(x-y)+b(d-x y)]$. The question is then can I not eliminate the variable $y$ from the second expression of the GCD i.e. $a(x-y)+b(d-x y)$ so that I am left with $GCD[(y^2-d)*(a^2-d b^2), (x^2-d)*(a^2-d b^2)]$? What is wrong in doing so?

John Bull
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    The gcds are not equal: if $,a=b=d=1,$ then the original gcd $= x-y+1-xy,$ but your derived gcd $= 0.\ $ You can use this to debug your proof line-by-line to find the error as explained here. – Bill Dubuque May 13 '23 at 21:35

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