How do you decide which side of a constraint line to shade out?

Test a point such as (0,0) in the inequality.

Where does the optimum of a linear objective lie?

At a vertex of the feasible region.

What is the objective line?

A line of constant objective value, e.g. 5x+4y=k, slid parallel across the region.

Which vertex does the objective line method choose when maximising?

The last vertex touched as the line moves away from the origin.

How do you find the vertex where two constraint lines meet?

Solve their equations simultaneously.

What does it mean if the objective line is parallel to a boundary edge?

Every point on that edge is optimal, so there are many optimal solutions.

Why is rounding an LP optimum risky for integer problems?

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

What is the method for an integer solution?

Find the LP optimum, then compare feasible whole-number points near it.

If the coefficients of P are whole numbers and x,y are whole numbers, what is P?

A whole number, so P\le19.5 means P\le19.

Vertex of 2x+y=12 and x+2y=10?

\left(\frac{14}{3},\frac83\right)

Graphical solution of linear programsEdexcel A-Level Further Maths: Flashcards

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What is the feasible region?

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What is the feasible region?: The set of points that satisfy all the constraints.
How do you decide which side of a constraint line to shade out?: Test a point such as $(0,0)$ in the inequality.
Where does the optimum of a linear objective lie?: At a vertex of the feasible region.
What is the objective line?: A line of constant objective value, e.g. $5x+4y=k$ , slid parallel across the region.
Gradient of $5x+4y=k$ ?: $-\frac54$
Which vertex does the objective line method choose when maximising?: The last vertex touched as the line moves away from the origin.
How do you find the vertex where two constraint lines meet?: Solve their equations simultaneously.
What does it mean if the objective line is parallel to a boundary edge?: Every point on that edge is optimal, so there are many optimal solutions.
Why is rounding an LP optimum risky for integer problems?: The rounded point may be infeasible or not the best integer point.
What is the method for an integer solution?: Find the LP optimum, then compare feasible whole-number points near it.
If the coefficients of $P$ are whole numbers and $x,y$ are whole numbers, what is $P$ ?: A whole number, so $P\le19.5$ means $P\le19$ .
Vertex of $2x+y=12$ and $x+2y=10$ ?: $\left(\frac{14}{3},\frac83\right)$

Exam questions on Graphical solution of linear programs

A cleaning company makes $x$ litres of cleaner A and $y$ litres of cleaner B each hour. The constraints are $2x+y\le12$ and $x+2y\le10$ , with $x\ge0$ and $y\ge0$ . The profit in pounds is $P=5x+4y$ . The quantities do not need to be whole numbers.
Find the maximum profit.2 marks
A zoo buys $x$ kg of feed X and $y$ kg of feed Y each week. The requirements are $x+y\ge8$ (total mass) and $2x+5y\ge30$ (protein units), with $x\ge0$ and $y\ge0$ . The weekly cost in pounds is $C=3x+4y$ , which the zoo wishes to minimise. Unless stated otherwise, feed can be bought in any quantity.
The zoo can now only buy whole numbers of kilograms of each feed. Find the minimum cost and the quantities that give it.2 marks
A joiner makes $x$ shelves and $y$ stools each week. The wood constraint is $4x+3y\le26$ and the time constraint is $x+2y\le10$ , with $x\ge0$ and $y\ge0$ . Each shelf gives £3 profit and each stool gives £2 profit, so the weekly profit is $P=3x+2y$ pounds.
Use the vertex method to find the maximum profit if shelves and stools did not need to be whole numbers.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).