Variables and Constants

In some situations, there can be quantities whose value can vary. Such quantities are represented using any letter of the English alphabet and are called variables. Example: x, y, p, q, a, b, c etc. This branch of Mathematics which deals with numbers and variables is called Algebra. Real life examples of variables include: the number of people visiting a library on different days, runs scored by a player in different matches, etc.

There are certain quantities which do not change their value; they remain fixed. Such fixed values are called constants. Example, numbers such as 4, 26, 100, etc. are constants.

• If a term has a variable, then it is called an ‘Algebraic term’.
• If the term is just a number, then it is called a ‘Constant term’.

Algebraic Expressions

Algebraic terms are terms with variables. They can take different values depending on the value of the variable. Constant terms are just numbers which have a fixed value. Terms combined with mathematical operators make an algebraic expression.

An algebraic expression is a mathematical statement containing variables. An algebraic expression can be described as a combination of constants and variables with four mathematical operations, namely addition, subtraction, multiplication and/or division.

Consider the following statement:

This can be written in a mathematical statement as:

Let us see a few more such statements.

Remember, numerical values in a statement are always fixed and constant.

Now let us learn how to form rules for different number patterns. Consider this pattern:

Let us make an observation.

If we consider the position of each number as ‘m’, then the rule will be:

So, we can form a rule for number patterns also. To form a rule for any given geometrical or number pattern:

• Identify the relation between two consecutive shapes or numbers.
• Form a relation between the position and the shape or number.
• Represent the changing value using a variable.
• Write the rule using the variable.

How do you write the correct rule matchstick patterns? Count the number of matchsticks in each shape of the pattern. Observe the position of the shape also.

Now, identify the pattern and its relationship with the position. Here, we see that in each case the number of matchsticks used is:

5 × position number + 1

Here, the position number of the shape is varying, hence, we represent it with a variable ‘n.’ The rule will be 5n + 1.

We can form algebraic expressions for given verbal statements involving different operations. Let us see how:

Addition

While converting a statement into an algebraic expression:

• Identify the unknown quantity in the statement.
• Identify the operation needed to relate the known and unknown quantities.
• ‘added’ or ‘sum of’ or ‘increased by’ or ‘more than’ – All mean addition.

Subtraction

When a statement is converted into an algebraic expression:

Identify the

unknown quantity.

Identify the operation needed to relate the known and unknown quantities.

‘Minus,’ ‘difference,’ ‘decreased by,’ ‘less than’ – All mean subtraction.

Multiplication

When a statement is converted into an algebraic expression:

• Identify the unknown quantity in the statement.
• Identify the operation needed to relate the known and unknown quantities.
• ‘Multiplied by,’ ‘times,’ ‘of’ – All mean multiplication.
• Sometimes you may have more than one operation involved.

Division

When a statement is converted into an algebraic expression:

• Identify the unknown quantity in the statement.
• Identify the operation needed to relate the known and unknown quantities.
• ‘Quotient,’ ‘divided by’ – All mean division.
• Sometimes you may have more than one operation involved.

Mathematical rules and formulae can be written as algebraic expressions. For example, the area of a rectangle is calculated as the product of the length and the breadth of the rectangle, i.e., area = length × breadth. We can write this formula as an algebraic expression by considering the length and the breadth of a rectangle as variables, namely ‘l’ and ‘b,’ respectively.

So, the formula to find the area of a rectangle can be written as:

Area = l × b = lb

For any shape such as a square, rectangle, triangle or circle, the perimeter is the distance around the shape. The perimeter can be written in algebraic form as the sum of all sides.

Algebraic expressions help us represent any rule as a general expression.

Now let us learn how to form an algebraic expression for commutative property of whole numbers. We know that when we add two whole numbers, the order in which we add them does not matter. In general, we can represent two whole numbers as ‘a’ and ‘b’.

Hence, according to the commutative property of addition for whole numbers, we have

Let us recall another important property of whole numbers. While multiplying three or more whole numbers, we can group the numbers and multiply. How we group the numbers does not really matter. So, in general, for any three whole numbers ‘a’, ‘b’ and ‘c’, we can state the associative property of multiplication as:

Similarly, we can represent the distributive property of whole numbers (‘a’, ‘b’ and ‘c’) as

Algebraic Equations

An algebraic equation has an ‘=’ sign with expressions or constants on either side of it. For e.g., 5x + 9 = 19. In this equation, we see that variables are also involved. In other words, algebraic terms are involved.

• The expression on the left of the ‘=’ is called the Left-Hand Side or LHS.
• The expression on the right of the ‘=’ is called Right-Hand Side or RHS.

An algebraic expression is just a combination of algebraic terms. There is no ‘=’ sign in it. For e.g., 4x + 6.

Can we form equations for a given mathematical statement in verbal form? To form an equation from a statement, we follow the steps given below:

• Identify the unknown and represent it with a variable.
• Identify the operation to be used.
• Look out for terms such as ‘added to’, ‘sum of’, more than’ or ‘increased’. These terms indicate that the operation to be used is addition (+).
• Form the equation.

What does the ‘equal to’ sign indicate in an equation? The equal sign indicates that both sides of the equation are equal, i.e., the equation is true only if the left- hand side expression is equal to the right-hand side expression.

Let us understand the meaning of satisfying the equation or solution of an equation. For example, consider the linear equation 5x + 12 = 27 where x = 3.

Let us take the value of ‘x’ to be 2.

So, we can conclude that x = 2 is not a solution to the equation 5x + 12 = 27.

This method of substituting different values and arriving at the value which satisfies the equation is called the Trial and Error method.

To check if a given value is a solution of an equation:

While solving real-life problems, we follow this strategy:

• Analyse the question to identify the known and unknown quantities.
• Form an equation relating the quantities.
• Solve the equation to find the value of the unknown.