Jacques I. Mathematics for Economics and Business

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After pressing the Enter key, the package will respond with the answer of 11. Try it now.

Notice that to get this answer, Maple must have performed the multiplication ﬁrst (to get 8)

before adding on the 3. This is because, like the rest of the mathematical world, Maple follows

the BIDMAS convention:

B (Brackets ﬁrst)

I (Indices second)

then D (Division)

and M (Multiplication)

(a tie for third place)

ﬁnally A (Addition)

and S (Subtraction)

(a tie for fourth place)

Since multiplication has a higher priority than addition, Maple works out 4 × 2 ﬁrst. If you

really want to work out 3 + 4 before multiplying by 2, you put in brackets:

>(3+4)*2;

which gives 14. Notice that in Maple (like Excel), it is necessary to use * rather than × as the

multiplication symbol. Similarly, Maple and Excel both use/(instead of ÷) for division, and ^

as the instruction to ‘raise to the power of ’. So, to work out + 5

you type:

>70/7+5^2;

which gives 35. Again, using BIDMAS, since indices have a higher priority than addition, 5

has

been evaluated ﬁrst as 25, before adding on 70/7 = 10, to give a ﬁnal answer of 35.

Introduction: Getting Started

Practice Problem

2 Use Maple to work out each of the following:

(a)

12 + 18 ÷ 9 (b) 3

− 4

MAPLE

Suppose now that you wish to work out all of the following sums:

3 × 5

− 2 × 5

3 × 6

− 2 × 6

3 × 7

− 2 × 7

3 × 6.4

− 2 × 6.4

3 × 92.5

− 2 × 92.5

You could, of course, just type all five calculations, one after the other to get the answers.

However, there is a common pattern. They each take the form

− 2x

for various values of x, and it makes sense to exploit this fact. As a first step, we shall give this

expression a name. We could call it Fred or Wilma, but in practice, we prefer to give it a name

that relates to the context in which it arises. A mathematical expression that contains a square

term like this is called a quadratic so let us name this particular one quad1. To do this, type:

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>quad1:=3*x^2-2*x;

The symbol ‘:=’ tells Maple that you wish to define quad1 to be 3x

− 2x.

To work out 3 × 5

− 2 × 5 we substitute x = 5 into this expression: that is, we replace the

symbol x by the number 5. In Maple, this is achieved by typing

>subs(x=5,quad1);

Maple responds with the answer 65. To perform the other four calculations, all we need do is

to edit the Maple instruction, change the 5 to a new value, move the cursor to the right of the

semi-colon, and press the Enter key. You might like to try this for yourself.

Hopefully, this brief introduction has given you some idea of how to use Maple to perform

simple calculations. However, you may be wondering what all the fuss is about. Surely we could

have performed these calculations just as easily on an ordinary calculator? Well, the honest

answer is probably yes. The real advantage of using Maple is as a tool for solving complex

mathematical problems. We shall meet some examples in the second half of this book. The

following problem gives a brief glimpse at the sort of things that it can do.

Introduction: Getting Started

Practice Problem

3 Type in the following after the ‘>’ prompt. Before doing so, you might like to guess what each

instruction is likely to do.

(a)

solve(2*x-8=0,x);

(b) plot(2*x-8,x=0..10);

(d) simplify(2*x+6+5*x-2);

(e) plot3d(x^3-3*x+x*y,x=-2..2,y=-2..2);

MAPLE

The key terms in each section are shown in colour in the text when they are ﬁrst used. Their

deﬁnitions are shown at the end of each section in a coloured box. In addition, all key terms

are repeated in the glossary at the end of the book.

Key Terms

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chapter 1

Linear Equations

The main aim of this chapter is to introduce the mathematics of linear equations.

This is an obvious first choice in an introductory text, since it is an easy topic which

has many applications. There are six sections, which are intended to be read in the

order that they appear.

Sections 1.1, 1.2, 1.4 and 1.5 are devoted to mathematical methods. They serve to

revise the rules of arithmetic and algebra, which you probably met at school but may

have forgotten. In particular, the properties of negative numbers and fractions are

considered. A reminder is given on how to multiply out brackets and how to manip-

ulate mathematical expressions. You are also shown how to solve simultaneous lin-

ear equations. Systems of two equations in two unknowns can be solved using

graphs, which are described in Section 1.1. However, the preferred method uses

elimination, which is considered in Section 1.2. This algebraic approach has the

advantage that it always gives an exact solution and it extends readily to larger sys-

tems of equations.

The remaining two sections are reserved for applications in microeconomics and

macroeconomics. You may be pleasantly surprised by how much economic theory

you can analyse using just the basic mathematical tools considered here. Section 1.3

introduces the fundamental concept of an economic function and describes how to

calculate equilibrium prices and quantities in supply and demand theory. Section 1.6

deals with national income determination in simple macroeconomic models.

The first five sections underpin the rest of the book and are essential reading. The

final section is not quite as important and can be omitted if desired.

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section 1.1

Graphs of linear equations

Consider the two straight lines shown in Figure 1.1. The horizontal line is referred to as the

x axis and the vertical line is referred to as the y axis. The point where these lines intersect is

known as the origin and is denoted by the letter O. These lines enable us to identify uniquely

any point, P, in terms of its coordinates (x, y). The ﬁrst number, x, denotes the horizontal

distance along the x axis and the second number, y, denotes the vertical distance along the

y axis. The arrows on the axes indicate the positive direction in each case.

Figure 1.1

Objectives

At the end of this section you should be able to:

 Plot points on graph paper given their coordinates.

 Add, subtract, multiply and divide negative numbers.

 Sketch a line by ﬁnding the coordinates of two points on the line.

 Solve simultaneous linear equations graphically.

 Sketch a line by using its slope and intercept.

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Linear Equations

Example

Plot the points A(2, 3), B(−1, 4), C(−3, −1), D(3, −2) and E(5, 0).

Solution

The point A with coordinates (2, 3) is obtained by starting at the origin, moving 2 units to the right and then

moving 3 units vertically upwards. Similarly, the point B with coordinates (−1, 4) is located 1 unit to the left

of O (because the x coordinate is negative) and 4 units up. These points, together with C(−3, −1), D(3, −2)

and E(5, 0) are plotted in Figure 1.2.

Note that the point C lies in the bottom left-hand quadrant since its x and y coordinates are both nega-

tive. It is also worth noticing that E actually lies on the x axis since its y coordinate is zero. Likewise, a point

with coordinates of the form (0, y) for some number y would lie somewhere on the y axis. Of course, the

point with coordinates (0, 0) is the origin, O.

Advice

The best way for you to understand mathematics is to practise the techniques yourself. For

this reason, problems are included within the text as well as at the end of every section.

Please stop reading the book, pick up a pencil and a ruler, and attempt these problems as

you go along. You should then check your answers honestly with those given at the back

of the book.



Practice Problem

1 Plot the following points on graph paper. What do you observe?

(2, 5), (1, 3), (0, 1), (−2, −3), (−3, −5)

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Before we can continue the discussion of graphs it is worthwhile revising the properties of

negative numbers. The rules for the multiplication of negative numbers are

×=

It does not matter in which order two numbers are multiplied, so

×=

These rules produce

(−2) × (−3) = 6

(−4) × 5 =−20

7 × (−5) =−35

respectively. Also, because division is the same sort of operation as multiplication (it just

undoes the result of multiplication and takes you back to where you started), exactly the same

rules apply when one number is divided by another. For example,

(−15) ÷ (−3) = 5

(−16) ÷ 2 =−8

2 ÷ (−4) =−1/2

In general, to multiply or divide lots of numbers it is probably simplest to ignore the signs to

begin with and just to work the answer out. The ﬁnal result is negative if the total number of

minus signs is odd and positive if the total number is even.

negative

negativepositive

negativepositivenegative

positivenegativenegative

1.1 • Graphs of linear equations

Figure 1.2

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Linear Equations

Example

Evaluate

(a) (−2) × (−4) × (−1) × 2 × (−1) × (−3)

(b)

Solution

(a) To evaluate

(−2) × (−4) × (−1) × (2) × (−1) × (−3)

we ignore the signs to get a preliminary value, 48. However, because there are an odd number of minus signs

altogether (in fact, ﬁve) the ﬁnal answer is −48.

(b) Ignoring the signs gives

==5

There are an even number of minus signs (in fact, four) so the answer is 5.

5 × 4 × 1 × 3

6 × 2

5 × (−4) × (−1) × (−3)

(−6) × 2

Advice

Attempt the following problem yourself both with and without a calculator. On most

machines a negative number such as −6 is entered by pressing the button labelled

followed by 6.

(−)

Practice Problem

(1) Without using a calculator evaluate

(a)

5 × (−6) (b) (−1) × (−2) (c) (−50) ÷ 10

(d) (−5) ÷ (−1) (e) 2 × (−1) × (−3) × 6 (f)

(2) Confirm your answer to part (1) using a calculator.

2 × (−1) × (−3) × 6

(−2) × 3 × 6



To add or subtract negative numbers it helps to think in terms of a picture of the x axis:

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If b is a positive number then

a − b

can be thought of as an instruction to start at a and to move b units to the left. For example,

1 − 3 =−2

because if you start at 1 and move 3 units to the left, you end up at −2:

Similarly,

−2 − 1 =−3

because 1 unit to the left of −2 is −3.

On the other hand,

a − (−b)

is taken to be a + b. This follows from the rule for multiplying two negative numbers, since

−(−b) = (−1) × (−b) = b

Consequently, to evaluate

a − (−b)

you start at a and move b units to the right (that is, in the positive direction). For example,

−2 − (−5) =−2 + 5 = 3

because if you start at −2 and move 5 units to the right you end up at 3.

1.1 • Graphs of linear equations

Example

Evaluate

(a) −32 − 4

(b) −68 − (−62)

Solution

(a) −32 − 4 =−36 because 4 units to the left of −32 is −36.

(b) −68 − (−62) =−68 + 62 =−6 because 62 units to the right of −68 is −6.

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