If −5 ≤ x ≤ 3 and −1 ≤ y ≤ 0, then the minimum value of 2y − 3x is:

Concept: A linear expression in variables that vary independently over a rectangular (box) domain reaches its extreme values at the corner points, and each term can be optimised on its own. A term with a positive coefficient is made as small as possible by taking its variable at the low end of its interval; a term with a negative coefficient is made as small as possible by taking its variable at the high end.

Application: Here the expression is f = 2y - 3x, and x and y vary independently within their given intervals.

1. The term 2y has a positive coefficient, so it is smallest when y is at its lowest value, y = -1, giving 2y = -2.

2. The term -3x has a negative coefficient, so it is smallest when x is at its highest value, x = 3, giving -3x = -9.

3. Adding the two smallest contributions: f = (-2) + (-9) = -11.

Cross-check: At the corner (x, y) = (3, -1): 2(-1) - 3(3) = -2 - 9 = -11. The opposite corner (x, y) = (-5, 0) gives 2(0) - 3(-5) = 15, which is the maximum, so -11 is indeed the minimum.