Linear programming: feasible region grapher
You don't have to test every point in the region — just the corners. So tilt the objective function and watch. The constraints here are fixed (a small feasible pentagon), but as you change the coefficients of Z = a·x + b·y, the dashed objective line swings, and a different corner lights up as the one that maximises Z.
The idea you're seeing
The feasible region
The set of every (x, y) point that satisfies all the constraints at once. For linear constraints this region is always a convex polygon — here, a pentagon with 5 corners.
The corner-point theorem
For a linear objective function over a bounded feasible region, the maximum (or minimum) value always occurs at a corner point — never strictly inside an edge or the interior.
Why corners, not edges
Sweep a straight objective-function line across the region and it always leaves the region last through a single vertex (or, in a tie, a whole edge) — never through an interior point.
Changing the objective changes the winner
Tilting the objective function — changing the ratio a : b — can make a different corner optimal. This is exactly what happens when real profits or costs per unit change.
Part of the Linear Programming chapter — read the notes, grab the formula sheet and take the quiz. One of Priodemy for School, free with every EduSuite school.
Why the answer is always at a corner
What the feasible region represents
Each constraint in a linear programming problem is an inequality, and each inequality shades half the plane. The feasible region is the overlap — the set of points that satisfy every constraint at once. In a real problem those points are the production plans you could actually carry out, given your limits on labour, raw material, machine hours or budget. Everything outside is impossible, not merely inefficient.
Because every constraint is linear, the boundary is made of straight edges and the region is a convex polygon. Convex means that a line joining any two feasible points stays inside the region. That property is what makes the whole method work, and it is why linear programming is reliable in a way that general optimisation is not.
The corner-point theorem
The objective function Z = ax + by is what you are maximising or minimising. Fix Z at some value and you get a straight line; change Z and the line moves parallel to itself, sweeping across the plane. Optimising means pushing that line as far as it will go while still touching the feasible region.
The last point of contact as the line leaves the region is a vertex. It cannot be somewhere in the middle, because you could always push the line further. This gives the corner-point theorem: if an optimum exists, it occurs at a vertex. The practical method follows directly — find the corners, evaluate Z at each, and compare. You never have to test the infinitely many interior points.
When two corners tie
Occasionally the objective line is exactly parallel to one of the edges. Then the two endpoints of that edge give the same value of Z, and so does every point along it — the problem has infinitely many optimal solutions. This is not an error or a trick question; it means the business genuinely has a range of equally good plans. Rotate the objective coefficients here until an edge lines up and you can watch the tie appear.
Mistakes that cost marks
Forgetting the non-negativity constraints. Almost every worded problem carries an unstated x ≥ 0 and y ≥ 0, because you cannot manufacture a negative number of chairs. Omitting them opens the region into quadrants that should not exist and usually produces a nonsensical optimum.
Reading corner coordinates off the graph instead of solving for them. A vertex is the intersection of two boundary lines, so solve those two equations simultaneously. Estimating from a drawing gives values like (3.9, 2.1) where the exact answer is (4, 2), and the marks are for the exact point.
Assuming an unbounded region has no answer. An unbounded feasible region may still have a minimum even though it has no maximum. Check which one the question actually asks for before concluding that no solution exists.