Class 6 — Mathematics (NCERT)

Introduction to Algebra: Variables, Expressions and Simple Equations

Class 6

✓By the end of this lesson students will be able to understand the concept of a variable and its use.
✓By the end of this lesson students will be able to form algebraic expressions from verbal statements.
✓By the end of this lesson students will be able to identify constants, variables, and operations in an algebraic expression.
✓By the end of this lesson students will be able to understand what an equation is and its components.
✓By the end of this lesson students will be able to solve simple equations using the trial and error method.

Key concepts

Variables

In mathematics, a variable is a quantity that can take various numerical values. It is usually represented by a letter of the English alphabet, such as x, y, z, a, b, m, p, etc. Variables help us represent unknown quantities or quantities that can change. For example, if we don't know the number of apples in a basket, we can represent it by 'x'.

Constants

A constant is a quantity that has a fixed numerical value. Unlike variables, its value does not change. For example, 5, 10, -3, 100 are all constants.

Algebraic Expressions

An algebraic expression is a combination of variables and constants, connected by some or all of the fundamental mathematical operations: addition (+), subtraction (-), multiplication (×), and division (÷). For example, x + 7, 3y - 2, 5m, and p/4 are all algebraic expressions.

Equations

An equation is a statement of equality between two algebraic expressions. It always contains an 'equal to' sign (=). The expression on the left side of the '=' sign is called the Left Hand Side (L.H.S.), and the expression on the right side is called the Right Hand Side (R.H.S.). For an equation to be true, the value of the L.H.S. must be equal to the value of the R.H.S. For example, x + 5 = 12, 2y = 10, and m - 3 = 7 are all equations.

Key facts to remember

1Algebra uses letters (variables) to represent unknown or changing numbers.
2A variable can take different numerical values, while a constant has a fixed numerical value.
3An algebraic expression is formed by combining variables and constants using mathematical operations.
4An equation is a statement of equality between two expressions, separated by an '=' sign.
5The Left Hand Side (L.H.S.) of an equation must be equal to its Right Hand Side (R.H.S.) for the equation to be true.
6The solution to an equation is the value of the variable that makes the equation true.

Worked examples

Example 1

Write the algebraic expression for the following statements:\n(a) 7 added to x\n(b) 5 subtracted from y\n(c) p multiplied by 4\n(d) m divided by 3

I(a) '7 added to x' means we add 7 to the variable x. So, the expression is x + 7.

II(b) '5 subtracted from y' means we subtract 5 from the variable y. So, the expression is y - 5.

III(c) 'p multiplied by 4' means we multiply the variable p by 4. This can be written as 4 × p or simply 4p.

IV(d) 'm divided by 3' means we divide the variable m by 3. This can be written as m ÷ 3 or m/3.

Answer

(a) x + 7\n(b) y - 5\n(c) 4p\n(d) m/3

Remember that 'subtracted from' means the number comes after the variable.

Example 2

Identify the variable, constants, and operations in the expression 3z + 8.

IStep 1: Identify the letter used. The letter 'z' represents a quantity that can vary, so 'z' is the variable.

IIStep 2: Identify the fixed numerical values. The numbers '3' and '8' have fixed values, so '3' and '8' are the constants.

IIIStep 3: Identify how the parts are connected. '3z' means 3 is multiplied by z. The '+' sign indicates addition. So, the operations are multiplication and addition.

Answer

Variable: z\nConstants: 3, 8\nOperations: Multiplication, Addition

Example 3

Solve the equation x + 6 = 10 using the trial and error method.

IStep 1: We need to find a value for 'x' such that when 6 is added to it, the result is 10. We will try different values for x.

IIStep 2: Let's try x = 1. L.H.S. = 1 + 6 = 7. R.H.S. = 10. Since 7 ≠ 10, x = 1 is not the solution.

IIIStep 3: Let's try x = 2. L.H.S. = 2 + 6 = 8. R.H.S. = 10. Since 8 ≠ 10, x = 2 is not the solution.

IVStep 4: Let's try x = 3. L.H.S. = 3 + 6 = 9. R.H.S. = 10. Since 9 ≠ 10, x = 3 is not the solution.

VStep 5: Let's try x = 4. L.H.S. = 4 + 6 = 10. R.H.S. = 10. Since L.H.S. = R.H.S., x = 4 is the correct solution.

Answer

x = 4

The trial and error method involves substituting different values for the variable until the L.H.S. equals the R.H.S.

Common mistakes

✗Confusing an expression with an equation: An equation always has an '=' sign, while an expression does not.
✗Incorrectly translating verbal statements into algebraic expressions, e.g., writing '5 less than x' as 5 - x instead of x - 5.
✗Forgetting that a number written next to a variable implies multiplication, e.g., 3x means 3 multiplied by x.
✗Not checking the solution by substituting the value of the variable back into the original equation.

Exam tips

★Always read the problem carefully to understand what is being asked and what quantities are unknown.
★Practice converting verbal statements into algebraic expressions to improve your understanding.
★When solving equations, clearly show each step of your working, especially for the trial and error method.
★After finding a solution to an equation, always substitute the value back into the original equation to verify your answer.

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