Question
Maximize subject to the constraints:
Find the maximum value of Z using the graphical method.
Solution — Step by Step
Take each constraint as an equation to draw boundary lines.
For : intercepts are and . For : intercepts are and .
Substitute the origin into each inequality to check feasibility — this is the fastest test.
✓ and ✓. So the origin lies in the feasible region, meaning we shade the side of each line that contains the origin.
The feasible region is bounded by four corner points. Three come directly from intercepts or the axes; the fourth is where our two boundary lines intersect.
Solve and simultaneously: Subtract the first from the second: , then .
Corner points: , , , .
This is the heart of the graphical method — the maximum (or minimum) of a linear objective function always occurs at a corner point of the feasible region.
| Corner Point | |
|---|---|
The highest value in the table is Z = 14 at (3, 1).
Maximum Z = 14, achieved at x = 3, y = 1.
Why This Works
A linear function on a convex polygon (which is what the feasible region always is) has no “hills” or “valleys” inside it. Every level curve is a straight line, and as we push higher, the line moves away from the origin until it just barely touches the feasible region. That last point of contact is always a corner.
This is why we never need to check interior points — they can never beat all the corners simultaneously. The Fundamental Theorem of Linear Programming guarantees this.
One geometric intuition worth building: the direction vector of is proportional to . The maximum is in whichever corner is furthest in that direction. Sketching the objective function line through the origin first — then sliding it in the direction — visually shows you which corner wins before you even calculate.
Alternative Method
Instead of evaluating Z at all four corners, we can use the objective function line method.
Draw for an arbitrary value, say (the line passes through the origin). Now slide this line parallel to itself, moving in the direction of increasing (away from origin, toward where both and grow). The last corner the line touches before exiting the feasible region is the maximum point.
This method is faster to execute on paper when the feasible region has many corner points, and CBSE occasionally asks you to show the “sliding line” approach for full marks in 4-mark questions.
When drawing the objective function line for the sliding method, choose a value of that gives a clean line. Here, passes through and — easy to draw and clearly outside the feasible region, which tells you immediately that you’ll be sliding inward.
Common Mistake
Missing the intersection corner point. The most common error in CBSE papers is listing only three corner points — the two axis intercepts of one line and the origin — and forgetting to find where the two constraint lines intersect each other. In this problem, is exactly that intersection point, and it gives the maximum. Students who skip it get Z = 12 instead of 14 and lose full marks. Always solve the simultaneous equations for every pair of boundary lines.