1. Variables 

• Consider the following ten balloons.

  
  

  

Can you divide these ten balloons into two groups in different ways? How many ways do you find when dividing these balloons into two groups?

Two approaches are given above for dividing these ten balloons into two groups and we have some more.

When we say that two numbers add up to 10, we cannot exactly say what these numbers are because there could be multiple possibilities that suit these two numbers. For an example, 

• 1 + 9 = 10

• 10 + 0 = 10

• 6 + 4 = 10

• 3 + 7 = 10

• 5.5 + 4.5 = 10
  

So now you can see that there are so many options when we think about two numbers that add up to 10. To resolve this issue, we can pick up two letters such as x and y so that these two letters represent two numbers that add up to 10.  

When x = 1, accordingly y = 9 since the sum of two numbers is 10. 

Similarly, 

when x = 2, y = 8.

when x = 3, y = 7

We can see that the value of x and y vary accordingly. They do not hold the same value. Therefore, x and y are called variables. Variable is a letter/symbol which is used to represent an unknown numerical value.

Now, if we consider the above situation, we can write an equation which makes two numbers add to 10.

x + y = 10

Similarly we can use variables to represent numbers when we do not know the exact values of those numbers. Since we initially do not know the values of the variables(letters), variables are also called unknown terms. 

2. Constants

A constant is a fixed value that does not change. Unlike variables which vary with a range of values based on the context, a constant always represents the same value.

Examples: 2, 3, -5, 0.2, \frac{5}{6}, ...

3. Algebraic terms and algebraic expressions

Mark wants to buy 3 ice cones but he does not know the price of one ice cone. Let $p be the price of one ice cone and let’s find the total price of 3 ice cones.

Since, one ice cone costs $p, we multiply p by 3 to find the total cost of all 3 ice cones. Total price = 3 × p 

Now, we can write the answer as 3p which represents the product of 3 and p. Therefore, total price = 3 × p = 3p. Moreover, here 3 which is followed by p is called the coefficient of p.

Here, 3p is called an algebraic term. 

Let’s consider a few more examples for algebraic terms.

• 5x → (5 \times x, coefficient of x is 5)

• -2y → (-2 \times y, coefficient of y is -2)

• \frac{a}{3} → (a \div 3, coefficient of a is \frac{1}{3})

• \frac{a}{2} → (a \div 2, coefficient of a is \frac{1}{2})

• 0.2r → (0.2 \times r, coefficient of a is 0.2)

  
  

• Now let’s consider the following question.

Write a statement for Jane’s father’s age, if Jane’s father is 7 years older than 3 times Jane’s age. Since Jane’s age is also unknown, let x be Jane’s age.

Now let’s workout 3 times Jane’s age, by multiplying Jane’s age which is x by 3. Therefore, 3 times Jane’s age is 3 × x = 3x. Then we can workout 7 more than 3 times Jane’s age by adding 7 to 3 times Jane’s age which is 3x. Therefore, 7 more than 3 times Jane’s age is 3x + 7.

Here, the statement that we got for Jane’s father’s age is called an algebraic expression. Hence, 3x + 7 is an expression. 

An algebraic expression is formed where algebraic terms and constants are combined with arithmetic operators such as addition, subtraction, multiplication and division. Let’s consider a few examples for algebraic expressions.

• x + 3

• 2y - 3

• \frac{a}{5} + 4

• x + y - 3
  

When multiplying an expression by a term or constant, we place the term or constant and the expression in parentheses next to each other to indicate their product. 

• x + 1 is multiplied by 2:

  2 \times (x + 1) = 2(x + 1)

• 2y - 3 is multiplied by (-5):

  -5 \times (2y - 3) = -5(2y - 3)

• (a - 10) is multiplied by 12:

  12 \times (a - 10) = 12(a - 10)
  

Construct an algebraic expression for each of the following situations

1. The price of a pineapple is $5 more than the price of 2 apples.

Let the price of one apple be $a. Then the price of a pineapple is = a \times 2 + 5 = 2a + 5

2. The width of a rectangle is 3 units less than its length. If the length is y units, what is the area in terms of y? 

Width = y - 3

Area = length × width

= y \times (y - 3)

= y(y - 3)

3. The height of a building is 6 meters less than \frac{1}{2} of its length. If its length is p meters, what is the height ?

Height = \frac{1}{2} \times p - 6 = \frac{p}{2} - 6 meters

4. Substitution

• Arnold wants to buy 3 apples and an orange. An orange costs $2 less than the price of an apple. 

Let’s write an expression for the price of an orange. Let x be the price of an apple.

Hence, the price of an orange is x - 2.

If we are given that the price of an apple is $5, we can find the price of an orange by replacing the value of xwith 5 in the expression x - 2. 

Therefore, when x - 5,

x - 2 = 5 - 2 = 3

Hence, the price of an orange is $3.

This process of replacing a variable in an expression with a specific value is called the substitution.

For example, let’s evaluate the expression 3y + 10 by substituting y = -2.

When y = -2; 

3y + 10 = 3 \times (-2) + 10

= -6 + 10

= 4

Examples

• Evaluate each of the following when a = 3