Session 1: Writing Expressions with Variables – Interactives and Examples

Nature is full of patterns. From music to art to architecture to medicine, many aspects of life involve patterns. If there is any pattern in mathematics, we use algebra to simplify and generalise this pattern.

Diophantus of Alexandria was a Greek mathematician. He is sometimes called “the Father of Algebra,” which he shares with Muhammad al-Khwarizmi. Diophantus represented his equations with words. In his book ‘Arithmetica’, he solved hundreds of algebraic equations and was the first to use algebraic notation and symbolism.  

In this unit, we will use algebra tools, such as writing and evaluating expressions, to explore patterns in the real world. Recognising a pattern and writing an expression that uses operations and numbers instead of words are essential algebra skills.

In algebra, we generally come across two types of symbols, namely constants and variables.

What is a Constant?

A symbol that has a fixed numerical value. Example, are all constants.

What is a Variable?

A symbol that can be assigned different numerical values or whose values can vary. It is interesting to note that the variables are known as literal coefficients. Literal means “letter”. Variables are generally represented by a letter of the English alphabet example: a,b,c, or x,y,z,etc. However, we avoid using e,i,and o because e and i stand for special mathematical terms and o to prevent confusion with zero.

Note: A combination of a constant and a variable, such as  are also variables as the value of these terms depend upon the value of x, which can be changed.

Operations on Literals and Constants

Since literals are used to represent numbers, the combination of literals and constants obeys all the rules and properties of addition, subtraction, multiplication, and division. 

Addition on literals and constants:

Word phrases for the following addition expression

For any literals x,y and z, we have:

Subtraction of Literals and Constants:

Word phrases for the following subtraction expression

Multiplication of Literals and Constants:

Word phrases for the following multiplication expression

The properties of multiplication of numbers also hold in algebra.

For any literals x,y  and z  we have

Division of Literals and Constants:

Word phases for the following division expression

Repeated addition of a Literal:

We have

 and so on.

Power of a Literal:

1. (read as x squared)
2.  (read as x cubed)
3.  (read as x to the power of four) and so on.

In general, we write

 taken n times = xⁿ

In xⁿ, we call x the base and n the exponent or index or power.

xⁿ is called the exponential form of a number.

We write,

Here is a list of some more algebraic statements. The signs and symbols used in these statements have the same meaning as they have in arithmetic.

• x is equal to y is x=y 
• x is not equal to y is x≠y 
• x is less than y is x<y 
• x is less than or equal to y is x≤y 
• x is greater than y is x>y 
• x is greater than or equal to y is x≥y 

Writing Expressions with Variables – Examples

Example1

Rewrite the following in the mathematical form in terms of variables and constants.

i. Twelve less than a number.

ii. Thrice a number added to 6.

  i. Let y represent the unknown number.
     Next, we look for the numbers in the phrase. The only number in the phrase is twelve.
     Next, identify the keywords in this phrase.
     “Less than” indicates subtraction.
     Now we write the phrase: 12 less than y 
     Since 12 is subtracted from y, the expression is written as y−12.

ii. Let x represent the unknown number.
    First, note that “thrice” is multiplication.   

 Next, identify the number in the phrase.

   6

   Now, identify the keywords and operations.
   “added” indicates the addition
   We rewrite the phrase as 3x added to 6 
   Since 3x is added to 6, the expression is written as 6+3x 

Example 2

Write each of the following in exponential form: 

i. We know that,

ii. We know that,

Example 3

Write each of the following in product form:

We have

Example 4

A wheel covers r centimetres in one revolution. How many centimetres does it cover in 75  complete revolutions?

We have,
Distance covered in one revolution =r centimetres
∴ Distance covered in 75 complete revolutions =75×r=75r  centimeters.

Example 5

In a classroom, there are 2x rows of benches. If each row has 3xy benches and each bench can accommodate x students, determine the number of students in the room if it is full up to its capacity.

We have, 
Number of rows in the room =2x 
Number of benches in each row =3xy

∴ Total number of benches in the room=                                                                 

Each bench can accommodate x students. 
∴ the total number of students = (number of benches) × (capacity of each bench)                                     

 Remember this! 

• An algebraic expression is a combination of constant and literals (variables) connected by the signs of fundamental operations. 
• A variable is a symbol that can be assigned different numerical values or whose values can vary. 
• A constant is a symbol that has a fixed numerical value.