BASIC PROCESSES OF ALGEBRA
You may have met the term ‘Algebra’ before. What do you understand by this term? Algebra
introduces the concept of variables representing numbers. In Algebra, letters are used as symbols
to represent numbers. Using these symbols, we can study the number relationships and the
properties of their operations.
In this unit we discuss the algebraic expressions and how they combine according the basic
arithmetical operations of addition, subtraction, multiplication and division.
Learning outcomes
As study and work through this unit you are expected to:
 use letters to represent symbols
 expand algebraic terms
 simplify algebraic expressions
Algebraic Expression
An algebraic expression is made up of the signs and symbols we use in algebra. These symbols
include Arabic numerals, literal numbers and signs of operation. Such an expression represents
one number or one quantity. Thus, just as the sum of 2 and3 is one quantity, that is 5, the sum of
a and b is one quantity, that is (a + b). Similarly, xy, x - y, and so forth, are algebraic expressions
each of which represents one quantity or number.
So let us move on and see we apply this concept.
Translating verbal phrases into algebraic expressions
By using letters, numbers and operation signs we can translate long verbal phrases into shorthand
algebraic expressions. For instance, how can you write the expression “the sum of an unknown
number and 5” in algebraic form?
Did you think of x  5 ? If you did, then you are correct. The letter x may be represented by
another letter, so the algebraic expression could be y  5, m + 5, j + 5 and so on.
Activity 10.1
Translate the following words into mathematical expressions
i. 5 less than a number
ii. subtract 5 from a number
iii. multiply a number by 3
iv. A number divided by 4
v. Double a number and add negative 3
Variables
A variable is a symbol, usually a letter, that represents one or more numbers. An expression that
contains one or more variables is an Algebraic expression. When you substitute numbers for
these variables and follow the order of operations, you are evaluating the expressions. An
expression is algebraic if at least one of its terms contains a letter. It is an arithmetic expression if
all the terms are numeric.
Examples
i. Algebraic Expressions;
a  2b,
cx  dy,
y2  2y  7
ii. Arithmetic Expressions;
2  4  7,
1
,
8
2 x 56
1
2x  5
Simplifying Algebraic Expressions
In an algebraic expression such as,  3x  4 , the parts that are added are called terms. A term is a
number, a variable or a product of a number and one or more variables. The numerical factor in a
term is called the numerical coefficient or simply coefficient. For instance, in the term,  3x ,
 3 is the coefficient and x is a variable.
Similarly, in the term, 2 y , 2 is a coefficient and y is a variable.
Activity 10.2
Like and Unlike Terms
Like terms have the same variable raised to the same power. Like terms are added or subtracted
by adding or subtracting the numerical coefficients and placing the result in front of the literal
factor, as in the following examples:
7x2 - 5x2 = (7 - 5)x2 = 2x2
5b2x - 3ay2 - 8b2x + 10ay2 = -3b2x + 7ay2
The expression
contains unlike terms. These cannot be added or subtracted to give
a single term.
Ways of Simplifying Algebraic Expressions
Let us look at a few examples of how to simplify algebraic expressions.
Examples
Simplify by combining the like terms;
a)
Remember we said like terms have the same variable raised to the same power. So 3k and k are
like terms. They have the same letter k raised to the same power 1. Therefore we may subtract k
from 3k as we have done above.
b)
Here, 5q2 and 8q2 are like terms (again look at the meaning of like terms). These may be added
or subtracted. Similarly 10q and q are like terms. They, too, may be added or subtracted
c)
Activity 10.3
1. Simplify the following;
i.
ii.
iii.
iv.
2. Find the perimeter
Distributive Law
For every real number
In order to find the value of
, you may first add the numbers in the brackets together
and then multiply the result by 2 or you may multiply 2 with each of the numbers in the brackets
and add the two products.
2(3+4) =2 (7) =14 OR 2(3+4) = (2
Examples
Expand and Simplify
a. 3(a + b)
b. 3(4x-2y) – 5(4y + x) +
c.
Solutions
a.
b. 3(4x-2y) – 5(4y + x) +
d.
Activity 10.4
1. Simplify each of the following expressions
i. 2x + 3y - 2 + 3x + 6y + 7
ii. 3b - (4b - 6b + 2) + b
iii.
iv.
v.
x + 2xy +3y +4x + 5y
vi. x2 + 5x + 4y + 7x + y2
2. Find the total cost of 4 books at x Kwacha each and 3 rulers at y Kwacha each.
3. Find the coefficient of xy in the expansion of
i. 3x  y a  2 y 
ii.
iii.
x  y 3x  2 y 
4x  2 y 5x  3 y 
iv.  x  3y 
2
Reflection
It is now time for you to reflect on what you have learned in this unit. Write down in your diary
the things that you think have added to your knowledge in the teaching of shapes to primary
school children or those aspects in which you used to experience difficulties but have been made
easier after studying through this unit.