Question

Find the maximization of \( z = x + y \), under the following constraints: \[ x - y \leq -1, -x + y \leq 0, x \geq 0, y \geq 0. \]

Hint:

In linear programming problems, solving the system of constraints often gives the vertices of the feasible region, where the maximum or minimum values occur.

Answer 1

We need to maximize \( z = x + y \), subject to the constraints: \[ x - y \leq -1, -x + y \leq 0, x \geq 0, y \geq 0. \] We begin by graphing the constraints: 1. \( x - y \leq -1 \) represents a line with slope 1 and intercept -1. 2. \( -x + y \leq 0 \) represents a line with slope 1 and intercept 0. 3. \( x \geq 0 \) is the region to the right of the y-axis. 4. \( y \geq 0 \) is the region above the x-axis. The feasible region is the area bounded by these lines. We can find the vertices of this region by solving the system of equations: 1. Solve \( x - y = -1 \) and \( -x + y = 0 \): \[ x - y = -1 \text{(i)} \] \[ -x + y = 0 \text{(ii)} \] From (ii), \( x = y \). Substituting into (i): \[ x - x = -1 $\Rightarrow$ 0 = -1, \] which is a contradiction, meaning there is no solution. Next, solve for other pairs of constraints to determine the vertices.