Question

Maximize \( z = x + y \) subject to: \[ x + y \leq 10, \quad 3y - 2x \leq 15, \quad x \leq 6, \quad x, y \geq 0. \] Find the maximum value.

Hint:

When maximizing or minimizing a linear objective function subject to linear constraints, the optimal value occurs at one of the vertices of the feasible region.

Answer 1

Step 1: Define the constraints for \(x\) and \(y\). The given constraints are: \( x + y \leq 10, \quad 3y - 2x \leq 15, \quad x \leq 6, \quad x, y \geq 0. \). These inequalities delineate the feasible region on a graph.

Step 2: Determine the vertices of the feasible region. Plotting these constraints graphically reveals the vertices at the intersections of the constraint lines. The vertices identified are \( (0, 0), (0, 5), (6, 4), (6, 0) \).

Step 3: Calculate the objective function's value at each vertex. The objective function is \( z = x + y \). Its values at the vertices are as follows:
- For \( (0, 0), z = 0 + 0 = 0 \).
- For \( (0, 5), z = 0 + 5 = 5 \).
- For \( (6, 4), z = 6 + 4 = 10 \).
- For \( (6, 0), z = 6 + 0 = 6 \).


Step 4: Conclude the maximum value. The maximum value of \( z \) is 10, occurring at the point \( (6, 4) \).