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Solve the system of equalities and inequalities

down vote

Backsubstitution is an option often ignored when people use Reduce. In the op’s case it gives:

Reduce[a == b + c && a >= 2 && b <= 10 && c == 5, {a, b, c}, Backsubstitution -> True]
2 <= a <= 15 && b == -5 + a && c == 5

via equation solving – Solve the system of equalities and inequalities – Mathematica Stack Exchange