How are the coordinates of a vertex found exactly?

Solve the equations of the two boundary lines that meet there.

State the vertex method.

Find all vertices, evaluate the objective function at each, choose the largest (or smallest).

Ruler method for maximising: which point is the optimum?

The last point of the feasible region touched when sliding the ruler parallel to the objective line away from the origin.

Ruler method for minimising: which point is the optimum?

The first point of the feasible region touched when sliding the ruler away from the origin.

In which direction does the ruler move?

Parallel to the objective line, never perpendicular.

What type of constraints do minimisation problems often have?

\ge constraints (requirements), so the region lies away from the origin.

Where does the optimum of a linear program lie?

At a vertex of the feasible region (or along an edge between two equally good vertices).

What must you state at the end?

The values of both variables and the optimal value of the objective function.

What is an integer solution?

A feasible point where every variable is a whole number.

How do you find the optimal integer solution?

Test integer points near the optimal vertex (or move the objective line back) checking every constraint.

Why is rounding the optimal vertex unreliable?

The rounded point may be infeasible or not the best integer point.

Graphical solution of linear programsEdexcel International A Level Maths: Flashcards

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What is the feasible region?

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What is the feasible region?: The set of points satisfying every constraint.
How are the coordinates of a vertex found exactly?: Solve the equations of the two boundary lines that meet there.
State the vertex method.: Find all vertices, evaluate the objective function at each, choose the largest (or smallest).
What is the gradient of the objective line $ax+by=k$ ?: $-\frac{a}{b}$
Ruler method for maximising: which point is the optimum?: The last point of the feasible region touched when sliding the ruler parallel to the objective line away from the origin.
Ruler method for minimising: which point is the optimum?: The first point of the feasible region touched when sliding the ruler away from the origin.
In which direction does the ruler move?: Parallel to the objective line, never perpendicular.
What type of constraints do minimisation problems often have?: $\ge$ constraints (requirements), so the region lies away from the origin.
Where does the optimum of a linear program lie?: At a vertex of the feasible region (or along an edge between two equally good vertices).
What must you state at the end?: The values of both variables and the optimal value of the objective function.
What is an integer solution?: A feasible point where every variable is a whole number.
How do you find the optimal integer solution?: Test integer points near the optimal vertex (or move the objective line back) checking every constraint.
Why is rounding the optimal vertex unreliable?: The rounded point may be infeasible or not the best integer point.
Is the best integer value ever better than the non-integer optimum?: No; it is equal or worse.

Exam questions on Graphical solution of linear programs

A furniture company makes $x$ desks and $y$ chairs each day. The company wants to maximise its daily profit $P=5x+4y$ (in hundreds of pounds), subject to the constraints $x+y\le10$ (worker hours), $3x+y\le18$ (workshop space), $x\ge0$ and $y\ge0$ .
The feasible region has vertices $(0,0)$ , $(6,0)$ , $(4,6)$ and $(0,10)$ . Use the vertex method to find the maximum value of $P$ and the values of $x$ and $y$ at which it occurs.2 marks
A school canteen mixes $x$ kg of ingredient X and $y$ kg of ingredient Y in each batch. The cost is $C=4x+3y$ pence, which is to be minimised, subject to $2x+y\ge12$ (energy), $x+2y\ge9$ (protein), $x+y\le10$ (tank capacity), $x\ge0$ and $y\ge0$ .
Show that the lines $2x+y=12$ and $x+2y=9$ meet at $(5,2)$ , and find the value of $C$ there.2 marks
A potter makes $x$ mugs and $y$ bowls each day. The clay available gives $2x+3y\le14$ and the kiln time available gives $4x+y\le15$ , with $x\ge0$ and $y\ge0$ . The profit is $P=5x+4y$ pounds, which is to be maximised.
Show that the lines $2x+3y=14$ and $4x+y=15$ meet at $(3.1,\,2.6)$ , and find the value of $P$ there.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).