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

Concept

A linear expression in two independent variables, each confined to a closed interval, attains its extreme values only at the endpoints (corners) of those intervals. To make a sum extreme, treat each term separately: for a MINIMUM, push each term as small (as negative) as possible, respecting the sign of its coefficient.

Application

Here the target is 2y - 3x, with x in [-5, 1] and y in [-1, 5]. Minimise each term:

1. Term 2y: the coefficient of y is positive, so 2y is smallest when y is smallest. Take y = -1, giving 2y = -2.

2. Term -3x: the coefficient of x is negative, so -3x is smallest when x is largest. Take x = 1, giving -3x = -3.

3. Add the two minimised terms: 2y - 3x = -2 + (-3) = -5.

Cross-check

Evaluate 2y - 3x at all four corners (x, y):

• (-5, -1): 2(-1) - 3(-5) = -2 + 15 = 13

• (-5, 5): 2(5) - 3(-5) = 10 + 15 = 25

• (1, -1): 2(-1) - 3(1) = -2 - 3 = -5

• (1, 5): 2(5) - 3(1) = 10 - 3 = 7

The smallest of these is -5, attained at x = 1, y = -1 — confirming the result.