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FM Graphs and Relations

2.5 Integer Solutions for Linear Programming Problems

Note: if you cannot remember how to analyse linear programming problems using graphical methods, revise notes for 2.4 Graphical Method for Solving Linear Programming Problems.

Integer Solutions for Linear Programming Problems

  • When using graphical methods for solving linear programming problems, we are presented with a region which encompasses all possible solutions. If the decision variables are continuous any point within this region it corresponds to a feasible solution. However, if we are dealing with discrete variables, only a finite number of points within this region will correspond to values the variables can actually take on.
  • If the region is sufficiently small, it is possible for us to find all possible solutions for a linear programming problem with integer solutions.
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2.4 Graphical Method for Solving Linear Programming Problems

Note: if you cannot remember how to setup a linear programming problem, revise notes for 2.3 Linear Programming.

Sliding Line Method

  • By drawing lines on a graph representing the minimum/maximum conditions as allowed by the systems inequalities, we bound a region. This region is known as the feasible region.
  • Any point within the feasible region represents a possible solution to this system. However, in general, we do not want just any possible solution, we want the “best” solution, known as the optimal solution. Often this is the combination of decision variable values that produces a maximum or minimum allowed value from the objective function.
  • If we substitute a point from the feasible region into the objective function we get a possible value for the quantity being optimised. If we then equate the objective function to this value, we produce a line of the form where a, b and c are constants and x and y are the decision variables. If we do this for another point, we will produce another equation which is parallel to the one just derived with the only difference being a different c value shifting the line upwards or downwards.
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2.3 Linear Programming

Note: these notes cover the formulation and understanding of linear programming questions, methods for determining solutions to these problems are covered in notes for 2.4 Graphical Method for Solving Linear Programming Problems and 2.5 Integer Solutions for Linear Programming Problems.

Linear Programming and its Practical Applications

  • Linear programming describes a process whereby we wish to maximise the value of a particular function, known as the objective function.
  • The objective function is based on a number of variables, known as the decision variables. In the scope of Further Maths, we will analyse situations with two decision variables.
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2.2 Graphs of Linear Inequalities

Graphs of Single Variable Inequalities

  • Inequalities can be represented graphically as a shaded area.
  • For single variable inequalities, the shaded area is enclosed by two vertical (for x inequalities) or horizontal (for y inequalities) lines.


Below is a graph representing the inequality -2 \leq x \leq 1:

Picture 1

Graphs of Two-Variable Inequalities

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2.1 Linear Inequalities

Single Variable Inequalities

  • Inequalities provide a range of values that a variable can possess.
  • Inequalities of the form x>a or x<a where x is a variable and a is a constant tell us that x is greater than a or less than a, respectively.
  • Adding a line beneath the inequality symbols indicates that the variable can also be equal to the value it is being equated to e.g. x \geq a means x is greater than or equal to a.
  • Inequalities can also be used to represent a finite range of values using the form:


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1.7 Linearisation Using Exponentials

Note: if you cannot remember the appearance of graphs of the form y=kx^{n}, revise notes for 1.6 Exponential Graphs.

Linearisation using x^{-1}

  • If a scatterplot appears to show a relationship of the form y=kx^{-1}, it can be linearised by taking the inverse of the data points.


















A company running a streaming service records the proportion of first-time users which return to the service over a period of 7 months, the results are plotted above. They require the data to be linearised in order to perform further analysis. From the dot plot, we can see that the relationship appears to be of the form: y=kx^{-1} and so we will take the inverse of the x data in order to linearise it:

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1.6 Exponential Graphs

Graphs of y=kxn

  • In Further Maths, we will analyse graphs of the form y=kx^{n}, where k is a constant known as the constant of proportionality, for the following values of n: -2, -1, 1, 2, 3. These forms are detailed individually below.

n= -2

  • The graph of y=kx^{-2} is symmetrical about the line x=0 and has two asymptotes. One at x=0, and one at y=0.
  • At x=0, the graph is undefined and on either side the graph curves upwards and goes towards infinity.
  • As you move away from the origin, the graph curves inwards and the value of y gets closer and closer to 0 the further you are from the origin.
  • All values of y are positive if k is positive, or negative if k is negative.


The graph of y=3x^{-2} is shown below:

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1.5 Use of Data and Graphs

Constructing Graphs from Tables

  • Given a table containing the x and y values of several points on a graph, we can construct a graph by plotting the known points and draw between them, using the trend of points to guess the overall shape of the graph.
  • When guessing the shape of the graph, there are many possible options. It is best to go with the simplest option.




















We wish to construct a graph from the above table. To begin, we plot the points on a graph:

Picture 2

Now, we draw between these points. In this case, the overall trend appears to curve and so we will draw a curved line:

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1.4 Analysing Non-Linear Graphs

Intercepts of Non-Linear Graphs

  • Non-linear graphs can have no intercepts, one intercept or multiple intercepts along either axis.


Picture 2

Above is a graph showing the profits earned by a community theatre in the years before they changed management. In the context of this situation, we can say that the intercepts at t=-3, -2 and 1 years represent points at which the community theatre just broke even (i.e. no profit or loss). The intercept at P=$1000 represents the profits made in the year the new management took over.

Maximum/Minimum Points

  • It may interest us to know the highest, or lowest values that the y-variable has taken. For example, if we want to find the highest a tide has been in a year in order to know how high to build a support structure.
  • The maximum and minimum points of non-linear graphs will be points of 0 gradient or the endpoints of a section (i.e. where a graph ends or undergoes structural change).
  • There may be multiple minimum points and/or multiple maximum points.
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1.3 Simultaneous Linear Equations

Systems of Simultaneous Linear Equations

  • To solve all unknown values in a set of simultaneous linear equations, the number of unknowns must be the same as the number of unique equations in the set.

Number of Solutions for Simultaneous Linear Equations

For two simultaneous linear equations

y=m_{1} x+c_{1}

y=m_{1} x+c_{1}

The number of solutions is defined by the gradient and y-intercept of the two equations.

Unique Solution (m_{1} \neq m_{2})

        • There is a single value for each variable that satisfies every equation.

No Solutions (m_{1} = m_{2}, and c_{1} \neq c_{2})

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