Question

Solve the following linear programming problem graphically: Maximise and minimise \( Z = 4x + 2y - 7 \) subject to the constraints: \[ x + 3y \leq 60, \quad x + y \geq 10, \quad x - y \leq 0, \quad x \geq 0, \quad y \geq 0 \]

Hint:

In graphical solutions to linear programming problems, always plot the constraints as equations, find the feasible region, and evaluate the objective function at each vertex.

Answer 1

Step 1: Graphing the constraints. 
To solve the linear programming problem graphically, we need to plot the constraints on the coordinate plane. First, convert the inequalities into equations: 
\( x + 3y = 60 \) 

\( x + y = 10 \) 

\( x - y = 0 \) 

\( x = 0 \) (the y-axis) 

\( y = 0 \) (the x-axis) 

Next, plot these lines on a graph. 
Step 2: Finding the feasible region. 
The feasible region is the region that satisfies all the inequalities. It is the intersection of all the half-planes determined by the inequalities. Identify the vertices of the feasible region. 
Step 3: Evaluating the objective function at the vertices. 
The objective function is \( Z = 4x + 2y - 7 \). Evaluate this function at each vertex of the feasible region. 
Step 4: Conclusion. 
The maximum and minimum values of \( Z \) occur at the vertices of the feasible region. The graphical solution will give the values of \( x \) and \( y \) that maximize and minimize \( Z \).