ALGEBRAIC EXPRESSIONS

BACKGROUND KNOWLEDGE

Algebra is an extension of arithmetic. In algebra,

in addition to use of numbers, letters e.g a, b, x, y e.t.c

are used to represent numbers.

The letters are handled in the same way numbers are

treated in arithmetic. Such an expression, where letters

are used to represent numbers is called an algebraic expression.

The following are some examples of algebraic expressions.

3a+b

4 +x

ab+c

7x-3y+2, a²+3

Example 1

Express the following statements in algebraic form.

a). The number of days in y weeks

Solution

One week has 7 days

Therefore y weeks will have y x 7 days = 7 days

b). Out of shs. 100 pocket money, a student spent shs x on books. 

Write an expression to represent the balance

Solution

Pocket money is shs. 100

Expenditure on books is shs. x

Balance is shs (100-x)

c). John bought x sweets and divided them to his 3 children 

equally. How many sweets did each child get.

Number of sweets are x

Number of children are 3

Number of sweets each child got is x/3 sweets

d). The cost of a pen and a pencil are shs. 10 and shs. 5 respectively.

Jane bought k pens and m pencils, while Mary bought 3 pens and 2 pencils.

How much did they spend altogether.

Cost of a pen is shs. 10 and of a pencil is shs. 5. Jane bought k pens

and m pencils. She spent shs. 10 x k = shs. 10k on pens and shs. 5 x m

on pencils. Her total expenditure = shs. 10k + shs. 5m

Mary bought 3 pens and paid shs. 3 x 10 = shs. 30

and bought 2 pencils paying shs. 2 x 5 = shs. 10.

In total mary spent shs. 30 + 10 = shs. 40

Altogether they spent

shs (10k +3m +40)

In any algebraic expression, parts connected by plus (+) or minus (-)

signs are called terms.

For example

10k +3m + 40 has terms, that is 10k, 3m and 40

In the term 10k, k has been multiplied with 10. 

10 is called the coefficient of k, and k the variable. In 3m, 3 is the

coefficient and m is the variable.

The term 40 has no variable, it is fixed, and is called constant

term in the expression.

Complete the following table

Expression

No of terms

List the variables

List the coefficients

State the constant term

3x+6y-10

3

x,y

3,6

10

4a-b+5

7-2c-6y+3y

+100

2x-y

Algebraic expressions have like and unlike terms. 

Terms are said to be like if they have exactly the same

variable(s) to the same power, otherwise they are unlike.

For example 6y and 9y are like

9xy and 2xy are like terms

2a2 and 6a2 are unlike terms

3a and 3b are unlike terms

4a and 6bc are unlike terms

3a and 3a2 are unlike terms

Worked out example 1

In each of the following pick out the like terms:

i). 2a, 5a, -7a, 2a2, 5b, 3a3

Solution

2a, 5a, -7a

ii). xy, 3x2y, 4, 4xy2, 2yx, 4xy

Solution

xy, 2yx, 4xy

iii). b2, 3b2, 7b, mn, 10b3, 2bmn

Solution

b2, 3b2

Algebraic expression can be simplified for example:

a). 2a + 3a + 5a

= 10a

b).7x + 4y + 2x + 3y

= 7x + 2x + 4y + 3y

= 9x + 7y

c). 3b + 9b - 4c + 7c

= 3b + 9b - 4c + 7c

= 12b +3c

In simplifying algebraic expressions first collect the

like terms together, and then perform the correct

operation involved.

Worked out examples 2

Simplify where possible

a). 7x - 2y + 4 - 3x + 7y

= 7x -3x - 2y + 7y +4

= 4x + 5y +4

b). 4c + 3d - 3c -d

= 4c - 3c + 3d - d

= 1c + 2d

c). 3a + 2ab - ac + 2ac - ad

= 3a + 2ab - ac + 2ac - ad

= 3a + 2ab + ac – ad

d). 48t – 7

e). 4mn2 + 7m2n - 6mn2 + 5m2n

= 4mn2-6mn2+7m2n+5m2n

= -2mn2 + 12m2n