Linear Programming: Maximize Profit

What is linear programming?

Linear programming finds the maximum or minimum of a linear objective function (like profit $P = 5x + 4y$) subject to linear constraints (like $x + y \leq 10$, $x \geq 0$).

The Corner Point Theorem

The optimal solution always occurs at a vertex (corner point) of the feasible region. So the algorithm is:

Step 1 — Graph the constraints to find the feasible region.

Step 2 — Find all corner points (where boundary lines intersect).

Step 3 — Evaluate the objective function at each corner.

Step 4 — The largest value is the maximum; the smallest is the minimum.

Step-by-step: Maximize $P = 5x + 4y$

Subject to: $x + y \leq 10$, $2x + y \leq 14$, $x \geq 0$, $y \geq 0$.

Corners: $(0,0)$, $(0,10)$, $(4,6)$, $(7,0)$.

$P(0,0) = 0$, $P(0,10) = 40$, $P(4,6) = 44$, $P(7,0) = 35$.

Maximum: $P = 44$ at $(4, 6)$.

Try it in the visualization

The feasible region is shaded. Corner points are marked with $P$ values. An objective function line sweeps to show visually which corner is optimal. Linear programming finds the maximum (or minimum) of a linear objective function subject to linear constraints. The optimal solution is always at a corner point (vertex) of the feasible region. The corner point theorem guarantees this.

Evaluate $P = 5x + 4y$ at each corner: $(0,0)$: $P=0$; $(0,10)$: $P=40$; $(4,6)$: $P=44$; $(7,0)$: $P=35$. Maximum is $P=44$ at $(4,6)$.