### Solving Linear Programming Problems In this section, we will address how to solve a given linear programming problem. #### Problem Statement: Minimize the objective function: \[ z = 7x + 26y \] Subject to the following constraints: 1. \( 4x + 3y \geq 21 \) 2. \( 4x + 8y \geq 32 \) 3. Non-negativity constraints: \( x \geq 0 \), \( y \geq 0 \) #### Objective: Determine the minimum value of \( z \). **Steps to Solve:** 1. **Graph the Constraints:** - Transform the inequality constraints into equations to identify boundary lines. - \( 4x + 3y = 21 \) - \( 4x + 8y = 32 \) - Plot these lines on a graph. - Identify the feasible region satisfying all inequality constraints. 2. **Find Corner Points:** - Identify the points of intersection for the lines and the coordinate axes. - For intersection of \( 4x + 3y = 21 \) and \( 4x + 8y = 32 \), solve simultaneously. - Find where each line intersects the axes. 3. **Evaluate the Objective Function:** - Calculate the value of \( z \) at each of the corner points identified in the feasible region. - Choose the point that gives the minimum \( z \). #### Example: - Intersection of \( 4x + 3y = 21 \) and \( 4x + 8y = 32 \): - Solve algebraically to find \( x \) and \( y \). - Intersection with axes: - For \( 4x + 3y = 21 \): - When \( x = 0 \), \( y = 7 \) - When \( y = 0 \), \( x = 5.25 \) - For \( 4x + 8y = 32 \): - When \( x = 0 \), \( y = 4 \) - When \( y = 0 \), \( x = 8 \) #### Solution Box: Once you find the point that gives the minimum value for the objective function: ``` What is the minimum value of z? z = [Type