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1. Operational Terms

Linear Programming deals with optimizing resources under specified constraint barriers.

Objective Function: Z = ax + by

This is the primary equation targeted for maximization (such as maximizing profit) or minimization (such as minimizing production costs).

Constraints: System of linear inequalities (e.g., x + y ≤ 5, 2x - y ≥ 0) along with non-negative restrictions (x ≥ 0, y ≥ 0) which force the solution space into the first quadrant.

2. Step-by-Step Graphing Sequence

Every constraint inequality must be converted into a boundary line equation and evaluated systematically:

{/* Reason: Skipping or misordering these step boundaries leads to shading the wrong half-plane, which completely breaks the solution. */} Turn the inequality into a standard equation: ax + by = c. Substitute x = 0 to calculate the y-intercept, and y = 0 to calculate the x-intercept. Substitute the coordinates of the origin (0,0) into the original inequality expression. If the result is True, shade the half-plane facing toward (0,0). If False, shade away from it. Plot all constraint lines on graph paper. Identify and darkly shade the overlapping, common intersection area. This intersection represents the Feasible Region. Identify the boundary vertices where the lines intersect to form the corners of the shaded region. Solve their intersecting equations simultaneously to find exact coordinate intersections.

3. Corner Point Optimization Method

According to optimization theory, the optimal maximum or minimum value will always occur at one of the corner vertices of the bounded space.

Required Optimization Evaluation Table:

Corner Vertices (x, y) Value of Objective Function: Z = ax + by Remarks / Evaluation
Origin Node (0, 0) Z = a(0) + b(0) = 0 Minimum Base Point
Vertex A (x₁, y₁) Z = a(x₁) + b(y₁) Calculated value
Vertex B (x₂, y₂) Z = a(x₂) + b(y₂) Calculated value