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I have a big system of equations and want to check if some candidate solutions hold.

Ideally, I want to have an expression that returns 'True' if my candidate solution hold and 'False' otherwise.

Based on previous posts, I decided to use Reduce[my_equtions && candidate_solutions && assumptions_on_variables]

However, Mathematica doesn't fully reduce the system to True / False -- even after using FullSimplify, some trivial terms remain in the output. Especially when my_equations are multiple, long equation, it is hard to find spot manually if the output should be true or not.

Consider as a MWE the system $x a^2 + y b^2 + (-x - y) ab == 0$ with the assumption $y>x>1$ over the reals. We have candidate solutions $a=1/x$, $b=1/y$ , which clearly solve the system.

However,

Reduce[x a^2 + y b^2 + (-x - y) ab == 0 &&  a == 1/x && b == 1/y && 
   y > x > 1, {a, b}, Reals] // FullSimplify

Outputs:

1 < x < y && ab == 1/(x y) && a == 1/x && b == 1/y

Ideally, I would get True or candidate_solutions && assumptions_on_variables. In any case, the part

ab == 1/(x y)

should not be in there, as these terms become very annoying when the system of equations is large.

My questions:

  1. How can I make Reduce to output simply True or candidate_solutions && assumptions_on_variables

  2. Is there a better command to test if given candidate solutions fulfill a system of equations under certain assumptions?

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    $\begingroup$ Your ab is not the same as a*b. Typing ab without space is a separate symbol. $\endgroup$
    – user293787
    Jul 26, 2022 at 18:03
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    $\begingroup$ The canonical way to check a candidate solution is eq=x*a^2+y*b^2+(-x-y)*a*b;candidate={a->1/x,b->1/y};eq/. candidate//Simplify. Another useful command in this ballpark is PossibleZeroQ. $\endgroup$
    – user293787
    Jul 26, 2022 at 18:05

2 Answers 2

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Simplify[
 x a^2 + y b^2 + (-x - y) a*b == 0, {a == 1/x, b == 1/y, y > x > 1}]

True.

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Provide the conditions to FullSimplify too.

eq = x a^2 + y b^2 + (-x - y) a b == 0;

cond = a == 1/x && b == 1/y && y > x > 1;

Reduce[eq && cond, {a, b}, Reals] // FullSimplify[#, cond] &

(*   True   *)
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