Chapter 9
Basic algebra
9.1 Introduction
We are already familiar with evaluating formulae using
a calculator from Chapter 4.
For example, if the length of a football pitch is L and its
width is b, then the formula for the area A is given by
A = L × b
This is an algebraic equation.
If L = 120 m and b = 60 m, then the area
A = 120 × 60 = 7200 m 2 .
The total resistance, R T , of resistors R 1 , R 2 and R 3
connected in series is given by
R T = R 1 + R 2 + R 3
This is an algebraic equation.
If R 1 = 6.3 k, R 2 = 2.4 k and R 3 = 8.5 k, then
R T = 6.3 + 2.4 + 8.5 = 17.2 k
The temperature in Fahrenheit, F, is given by
F =
9
5
C + 32
where C is the temperature in Celsius. This is an
algebraic equation.
If C = 100 ◦ C, then F =
9
5
× 100 + 32
= 180 + 32 = 212 ◦ F.
If you can cope with evaluating formulae then you will
be able to cope with algebra.
9.2 Basic operations
Algebra merely uses letters to represent numbers.
If, say, a, b, c and d represent any four numbers then in
algebra:
(a) a + a + a + a = 4a. For example, if a = 2, then
2 + 2 + 2 + 2 = 4 × 2 = 8.
(b) 5b means 5 × b. For example, if b = 4, then
5b = 5 × 4 = 20.
(c) 2a + 3b + a − 2b = 2a + a + 3b − 2b = 3a + b
Only similar terms can be combined in algebra.
The 2a and the +a can be combined to give 3a
and the 3b and −2b can be combined to give 1b,
which is written as b.
In addition, with terms separated by + and − signs,
the order in which they are written does not matter.
In this example, 2a + 3b + a − 2b is the same as
2a + a + 3b − 2b, which is the same as 3b + a +
2a − 2b, and so on. (Note that the first term, i.e.
2a, means +2a.)
(d) 4abcd = 4 × a × b × c × d
For example, if a = 3, b = −2, c = 1 and d = −5,
then 4abcd = 4 × 3 × −2 × 1 × −5 = 120. (Note
that − × − = +)
(e) (a)(c)(d) means a × c × d
Brackets are often used instead of multiplication
signs. For example, (2)(5)(3) means 2 × 5 × 3 =
30.
(f ) ab = ba
If a = 2 and b = 3 then 2 × 3 is exactly the same
as 3 × 2, i.e. 6.
(g) b
2
= b × b. For example, if b = 3, then
3 2 = 3 × 3 = 9.
(h) a 3 = a × a × a For example, if a = 2, then
2 3 = 2 × 2 × 2 = 8.
Here are some worked examples to help get a feel for
basic operations in this introduction to algebra.
DOI: 10.1016/B978-1-85617-697-2.00009-0
Basic algebra
9.1 Introduction
We are already familiar with evaluating formulae using
a calculator from Chapter 4.
For example, if the length of a football pitch is L and its
width is b, then the formula for the area A is given by
A = L × b
This is an algebraic equation.
If L = 120 m and b = 60 m, then the area
A = 120 × 60 = 7200 m 2 .
The total resistance, R T , of resistors R 1 , R 2 and R 3
connected in series is given by
R T = R 1 + R 2 + R 3
This is an algebraic equation.
If R 1 = 6.3 k, R 2 = 2.4 k and R 3 = 8.5 k, then
R T = 6.3 + 2.4 + 8.5 = 17.2 k
The temperature in Fahrenheit, F, is given by
F =
9
5
C + 32
where C is the temperature in Celsius. This is an
algebraic equation.
If C = 100 ◦ C, then F =
9
5
× 100 + 32
= 180 + 32 = 212 ◦ F.
If you can cope with evaluating formulae then you will
be able to cope with algebra.
9.2 Basic operations
Algebra merely uses letters to represent numbers.
If, say, a, b, c and d represent any four numbers then in
algebra:
(a) a + a + a + a = 4a. For example, if a = 2, then
2 + 2 + 2 + 2 = 4 × 2 = 8.
(b) 5b means 5 × b. For example, if b = 4, then
5b = 5 × 4 = 20.
(c) 2a + 3b + a − 2b = 2a + a + 3b − 2b = 3a + b
Only similar terms can be combined in algebra.
The 2a and the +a can be combined to give 3a
and the 3b and −2b can be combined to give 1b,
which is written as b.
In addition, with terms separated by + and − signs,
the order in which they are written does not matter.
In this example, 2a + 3b + a − 2b is the same as
2a + a + 3b − 2b, which is the same as 3b + a +
2a − 2b, and so on. (Note that the first term, i.e.
2a, means +2a.)
(d) 4abcd = 4 × a × b × c × d
For example, if a = 3, b = −2, c = 1 and d = −5,
then 4abcd = 4 × 3 × −2 × 1 × −5 = 120. (Note
that − × − = +)
(e) (a)(c)(d) means a × c × d
Brackets are often used instead of multiplication
signs. For example, (2)(5)(3) means 2 × 5 × 3 =
30.
(f ) ab = ba
If a = 2 and b = 3 then 2 × 3 is exactly the same
as 3 × 2, i.e. 6.
(g) b
2
= b × b. For example, if b = 3, then
3 2 = 3 × 3 = 9.
(h) a 3 = a × a × a For example, if a = 2, then
2 3 = 2 × 2 × 2 = 8.
Here are some worked examples to help get a feel for
basic operations in this introduction to algebra.
DOI: 10.1016/B978-1-85617-697-2.00009-0