Quick questions on Linear Programming - TCE Mathematics Applications (Tasmania)

Read the problem and define $x$ and $y$ precisely, including units. Each resource limit becomes one inequality. Almost every LP problem also includes the non-negativity constraints $x \ge 0$ and $y \ge 0$, because you cannot make a negative number of items.

Find the coordinates of each corner (vertex), often by solving the two boundary lines that meet there simultaneously. Substitute each corner into the objective function and compare the values.

An alternative is to draw the objective line $P = 60x + 40y$ for any convenient value of $P$, then slide it parallel to itself in the direction of increasing $P$. The last corner it touches before leaving the feasible region is the optimum. This gives the same answer as testing corners and is a good visual check.

The arithmetic that trips students up is finding where two boundary lines cross. Treat each binding constraint as an equation and solve the pair simultaneously, usually by elimination. For example, $2x + y = 40$ and $x + y = 30$ subtract to give $x = 10$, and back-substitution gives $y = 20$. Corners on an axis are easier: set $x = 0$ or $y = 0$ in the relevant boundary line.