For linear programming, maximum occurs at vertices of the feasible region.
The problem requires maximizing the objective function \( z = 5x + 2y \) under the constraints \( 2x + y \leq 8 \), with \( x \geq 0 \) and \( y \geq 0 \).
Step 1: Define constraints and identify the feasible region.
The constraint \( 2x + y \leq 8 \) is represented by the line \( y = -2x + 8 \). The feasible region is enclosed by this line, the x-axis, and the y-axis, forming a triangular area with vertices at the origin (0,0), an x-intercept, and a y-intercept.
Step 2: Determine the intersection points with the coordinate axes.
For the x-axis (\(y = 0\)), solving \( 2x = 8 \) yields \( x = 4 \). The point is (4, 0).
For the y-axis (\(x = 0\)), \( y = 8 \). The point is (0, 8).
Step 3: Calculate the objective function's value at each vertex.
1. At (0, 0): \( z = 5(0) + 2(0) = 0 \).
2. At (4, 0): \( z = 5(4) + 2(0) = 20 \).
3. At (0, 8): \( z = 5(0) + 2(8) = 16 \).
The highest value of \( z \) among these vertices is 20.
Therefore, the maximum value \( z \) can attain within the defined feasible region is 20.
| Vertex | \((x, y)\) | Value of \(z\) |
| 1 | (0, 0) | 0 |
| 2 | (4, 0) | 20 |
| 3 | (0, 8) | 16 |