Class 6 CBSE Maths Notes: Algebra - Complete Guide with Formulas & Practice Questions
Algebra is a fundamental branch of mathematics introduced in Class 6 CBSE maths notes that teaches students to use letters (variables) to represent unknown numbers and solve real-world problems.
This chapter forms the foundation for advanced mathematical concepts in higher classes, making it essential for students to develop crystal-clear understanding of basic algebraic principles.
What is Algebra?
Algebra uses letters called variables (like x, y, z) along with constants and the four basic operations (addition, subtraction, multiplication, and division) to create general formulas and solve problems. For example, the formula for the area of a triangle A = 1/2 × b × h, where A, b, and h represent numbers that can vary depending on the specific triangle. When we translate real-world situations into mathematical expressions using variables, we are using algebra. If Sonu's current age is x years and her mother is 5 times her age, then her mother's age equals 5x years.
Algebraic Concepts for Class 6 Students
Variables and Constants
A variable is a letter or symbol that represents an unknown value that can change. Constants are fixed numerical values that do not change. In the expression 4x + 7, x is the variable while 4 and 7 are constants.
Algebraic Expressions
Algebraic expressions are combinations of variables, constants, and mathematical operations. They translate word statements into mathematical form.
- A number increased by 6: x + 6.
- A number decreased by 7: x - 7.
- 12th part of a number: x/12.
- 4 times a number: 4x.
- 2 more than 4 times a number: 4x + 2.
Types of Algebraic Expressions
Students learn about different expression types including monomials (single term), binomials (two terms), and trinomials (three terms).
Linear Equations in One Variable
A linear equation is an equation where the highest power of the variable is 1. The expression x + 8 = 15 is a linear equation in one variable.
A solution of an equation is a number that makes the left-hand side (LHS) equal to the right-hand side (RHS) when substituted for the variable.
Rules for Solving Linear Equations
Class 6 students learn four fundamental rules for solving equations.
- Addition Rule: Add the same number to both sides of an equation.
- Subtraction Rule: Subtract the same number from both sides.
- Multiplication Rule: Multiply both sides by the same non-zero number.
- Division Rule: Divide both sides by the same non-zero number.
Transposition Method
Transposition allows moving any term from one side of an equation to the other by changing its sign. For equation x + 8 = 3, 8 is transposed to the right side: x = 3 - 8, giving x = -5.
Essential Algebra Formulas and Identities for Class 6
| Formula Name | Mathematical Representation | Explanation |
|---|---|---|
| Area of Triangle | A = 1/2 x b x h | A is area, b is base, h is height. |
| Perimeter of Rectangle | P = 2 (l + b) | P is perimeter, l is length, b is breadth. |
| Cost Expression | Total Cost = nx | n is number of items, x is cost per item. |
| Age Problems | Present age + years = Future age | Basic form x + n where x is present age and n is years. |
| Consecutive Numbers | x, x + 1, x + 2... | Represents consecutive natural numbers. |
What Are the Main Branches of Algebra?
Algebra encompasses several specialized branches that students encounter as they progress in mathematics. Elementary algebra, introduced in Class 6, focuses on basic operations with variables and solving simple equations. Linear algebra studies linear equations, matrices, vector spaces, and linear transformations in finite and infinite dimensions.
Abstract algebra (or modern algebra) examines mathematical structures like groups, rings, fields, modules, and lattices, providing a generalized approach to mathematics.
The key distinction is that linear algebra concentrates on linear transformations where T (a + b) = T(a) + (b) and kT(a) = T(ka), while abstract algebra encompasses these and many other non-linear structures.
Important Concepts to Learn First for Beginner Algebra Students
Beginner algebra students should master foundational topics in a logical sequence. Basic arithmetic operations with integers, decimals, and fractions.
- Understanding variables as letters representing unknown values.
- Recognising constants as fixed numerical values.
- Writing algebraic expressions from word problems.
- Evaluating expressions by substituting values.
- Solving simple equations and finding their solutions.
- Using transposition to rearrange and solve equations.
- Translating real-life word problems into mathematical equations.
Students should develop comfort with integers, decimals, fractions, powers, and roots before advancing to more complex algebraic concepts.
How is Linear Algebra Different from Abstract Algebra?
Linear algebra and abstract algebra serve different but related mathematical purposes. Linear algebra focuses on linear equations, systems of linear equations, matrices, vector spaces, and linear transformations. It deals with problems that can be expressed in linear form and solved using matrix operations, with applications in economics, engineering, and physics.
Abstract algebra takes a more theoretical approach, studying general structures including groups, rings, fields, vector spaces, and lattices. Abstract algebra encompasses linear algebra as a special case but extends to a wide range of structures to establish mathematics on rigorous theoretical foundations.
Common Algebra Formulas and Identities to Memorize
Beyond Class 6 basics, students should become familiar with fundamental algebraic operations. [web:10]
- Addition: x + y.
- Subtraction: x - y.
- Multiplication: xy or x × y.
- Division: x/y or x ÷ y.
Operations follow the BODMAS rule (Brackets, Orders/Exponents, Division, Multiplication, Addition, Subtraction) for the proper order of execution.
Solved Examples from Class 6 Algebra
Example 1: Age Problem
Mrs. Goel is 27 years older than her daughter Rekha. After 8 years she will be twice as old as Rekha. Find their present ages.
Solution:
Let Rekha's present age = x years, then Mrs. Goel's present age = x + 27 years. After 8 years, Rekha's age becomes x + 8 and Mrs. Goel's age becomes x + 35. According to the condition, x + 35 = 2(x + 8), which simplifies to x = 19 years.
Therefore, Rekha is 19 years old and Mrs. Goel is 46 years old at present.
Example 2: Perimeter Problem
The length of a rectangular field is twice its breadth. If the perimeter is 228 m, find the dimensions.
Solution:
Let breadth = b, so length = 2b. Using perimeter formula 2(length + breadth) = 228 gives 2(2b + b) = 228, so 6b = 228 and b = 38 m.
Thus, breadth = 38 m and length = 76 m.
Example 3: Consecutive Numbers
The sum of two consecutive natural numbers is 21. Find the numbers.
Solution:
Let the numbers be x and x + 1. Then x + x + 1 = 21, so 2x + 1 = 21, which gives 2x = 20 and x = 10.
Hence, the two numbers are 10 and 11.
Practice Questions for CBSE Class 6 Algebra
Short Answer Questions
- Solve: x - 7 = 6.
- If a number is tripled and increased by 5, we get 50. Find the number.
- Translate: “Rajeev is 5 years older than double the age of Somya” into an algebraic expression.
Long Answer Questions
- The sum of three consecutive natural numbers is 114. Find the numbers.
- A man is 4 times as old as his son. After 16 years he will be twice as old as his son. Find their present ages.
- The length of a rectangular hall is 5 metres more than its breadth. If the perimeter is 74 metres, find the length and breadth.
Tips for Mastering Class 6 Algebra
- Practice converting word statements into algebraic expressions regularly.
- Master the transposition method for faster and cleaner equation solving.
- Verify every solution by substituting back into the original equation.
- Solve a variety of questions, including age problems, number problems, and geometry-related applications.
- Focus on conceptual understanding instead of memorising steps.
- Work through solved examples before attempting similar practice questions independently.
Frequently Asked Questions
A variable is a letter or symbol (like x, y, or z) that represents an unknown or changing value in algebraic expressions and equations.
An expression is a mathematical phrase containing variables, constants, and operations, whereas an equation has an equals sign showing two expressions are equal.
A linear equation in one variable is an equation where the highest power of the variable is 1, such as 2x + 7 = 15.
Simple linear equations are solved by adding, subtracting, multiplying, or dividing the same number on both sides, or by using transposition to move terms across the equals sign with sign change.
Transposition is the process of moving a term from one side of an equation to the other by changing its sign from positive to negative or vice versa, or from multiplication to division and vice versa.
To write algebraic expressions, identify the unknown quantity as a variable and translate words like increased, decreased, and times into the corresponding operations.
The solution of an equation is the value of the variable that makes the left-hand side equal to the right-hand side when substituted in the equation.
Constants are fixed numerical values that do not change, unlike variables; for example in 5x + 8, the numbers 5 and 8 are constants.
Substitute the value of the variable back into the original equation; if the left-hand side equals the right-hand side, the solution is correct.
Algebra builds the foundation for higher mathematics, develops logical problem-solving skills, and supports preparation for future competitive examinations.