CBSE Class 10 Maths Chapter 3 Linear Equations Notes (NCERT Solutions)

CBSE Class 10 Maths Notes – Chapter 3 Pair of Linear Equations in Two Variables

Complete NCERT solutions with formulas, methods, and board exam questions for 2025

Chapter 3: Pair of Linear Equations in Two Variables - Complete Study Material for CBSE Board Exams 2025

Welcome to CBSE Class 10 Notes! These notes are prepared according to NCERT syllabus and are very useful for CBSE Board Exams 2025. Here you will find all formulas, solved examples, and important questions.

Table of Contents

1. Introduction
2. Chapter Overview
3. Key Concepts
4. Methods of Solving
5. Smart Tricks
6. Important Board Questions
7. Word Problems
8. Common Mistakes
9. Practice MCQs
10. FAQ

1. Introduction

The general form of a linear equation in two variables is: ax + by + c = 0

Where a, b, c are real numbers and a and b cannot both be zero.

Chapter Importance:

• Weightage of 4-7 marks in Board Exams
• Word problems appear every year
• Important foundation for Class 11
• 100% NCERT aligned

2. Chapter Overview

Topic	Marks	Question Type
Graphical Method	2-3	Short Answer
Algebraic Methods	3-4	Long Answer
Word Problems	4-5	Long Answer
Consistency	1	MCQ

3. Key Concepts

General Form

ax + by + c = 0

Pair of Linear Equations

a₁x + b₁y + c₁ = 0

a₂x + b₂y + c₂ = 0

Conditions for Solutions

Condition	Geometric Meaning	Solution Type
a₁/a₂ ≠ b₁/b₂	Intersecting lines	Unique Solution
a₁/a₂ = b₁/b₂ ≠ c₁/c₂	Parallel lines	No Solution
a₁/a₂ = b₁/b₂ = c₁/c₂	Coincident lines	Infinite Solutions

Figure: Types of Solutions - Intersecting, Parallel and Coincident Lines

4. Methods of Solving

Method 1: Graphical Method

1. Write both equations in y = mx + c form

2. Find at least 3 points for each equation

3. Plot the points on graph paper

4. Draw lines and find intersection point

Example:

Solve graphically: x + y = 5 and 2x - y = 4

Solution:
Equation 1: y = 5 - x
Points: (0,5), (1,4), (2,3)

Equation 2: y = 2x - 4
Points: (0,-4), (2,0), (3,2)

Lines intersect at (3, 2).

Answer: x = 3, y = 2

Method 2: Substitution Method

Example:

Solve: 2x + y = 7 and x - y = -1

Solution:
From Equation 2: x = y - 1

Substitute in Equation 1:
2(y - 1) + y = 7
2y - 2 + y = 7
3y = 9
y = 3

x = 3 - 1 = 2

Answer: x = 2, y = 3

Method 3: Elimination Method

Example:

Solve: 3x + 2y = 12 and 2x - y = 3

Solution:
Multiply Equation 2 by 2:
4x - 2y = 6

Add to Equation 1:
(3x + 2y) + (4x - 2y) = 12 + 6
7x = 18
x = 18/7

y = 15/7

Answer: x = 18/7, y = 15/7

Method 4: Cross-Multiplication Method

Cross-Multiplication Formula:

x / (b₁c₂ - b₂c₁) = y / (c₁a₂ - c₂a₁) = 1 / (a₁b₂ - a₂b₁)

Example:

Solve: 2x + 3y = 8 and 4x - y = 2

Solution:
a₁=2, b₁=3, c₁=-8
a₂=4, b₂=-1, c₂=-2

x / [(3)(-2) - (-1)(-8)] = y / [(-8)(4) - (-2)(2)] = 1 / [(2)(-1) - (4)(3)]

x / (-14) = y / (-28) = 1 / (-14)

x = 1, y = 2

Answer: x = 1, y = 2

5. Smart Tricks

Trick 1: Fast Solution Type Check

First calculate a₁/a₂, b₁/b₂, c₁/c₂:
• If first two are different → Unique Solution
• If all three are equal → Infinite Solutions
• If first two equal but third different → No Solution

Trick 2: Determinant Check

If (a₁b₂ - a₂b₁) ≠ 0 → Unique Solution
If = 0 and (b₁c₂ - b₂c₁) ≠ 0 → No Solution

Trick 3: Time Saving

Always verify your answer by substituting in original equations. It takes 30 seconds and catches errors.

6. Important Board Questions

★ 1 Mark Questions

Q: For what value of k will the system have no solution?
2x + 3y = 7 and 4x + ky = 15

Answer: k = 6 (because 2/4 = 3/6 ≠ 7/15)

★ 2-3 Mark Questions

Q: Solve: 3x + 4y = 10 and 2x - 3y = 1

Using Elimination method:
Eq1 × 3: 9x + 12y = 30
Eq2 × 4: 8x - 12y = 4
Add: 17x = 34 → x = 2
y = 1

Answer: x = 2, y = 1

★★★ 4-5 Mark Word Problems

Age Problem (CBSE 2023):

Five years ago, A was thrice as old as B. Ten years later, A shall be twice as old as B. Find their present ages.

Solution:
Let present age of A = x
Let present age of B = y

Five years ago: x - 5 = 3(y - 5)
→ x - 3y = -10 ... (i)

Ten years later: x + 10 = 2(y + 10)
→ x - 2y = 10 ... (ii)

(ii) - (i): y = 20
x = 50

Answer: A = 50 years, B = 20 years

7. Word Problems

Number Problem

Q: The sum of digits of a two-digit number is 9. If 27 is added to the number, the digits interchange their places. Find the number.

Let unit digit = x, tens digit = y
Number = 10y + x

x + y = 9 ... (i)
10y + x + 27 = 10x + y
→ 9y - 9x = -27
→ y - x = -3 ... (ii)

(i) + (ii): 2y = 6 → y = 3
x = 6

Answer: 36

Speed-Distance Problem

Q: A boat goes 30 km upstream and 44 km downstream in 10 hours. It also goes 40 km upstream and 55 km downstream in 13 hours. Find the speed of the boat in still water.

Let speed of boat = x km/h
Let speed of stream = y km/h

30/(x-y) + 44/(x+y) = 10
40/(x-y) + 55/(x+y) = 13

Let 1/(x-y) = u and 1/(x+y) = v:
30u + 44v = 10
40u + 55v = 13

Solving: u = 1/5, v = 1/11
x - y = 5 and x + y = 11
x = 8, y = 3

Answer: Speed of boat = 8 km/h, Speed of stream = 3 km/h

8. Common Mistakes

1. Sign Errors:

Mistake in writing 2x - 3y = 7 as 2x + (-3y) = 7.
Fix: Always write in standard form ax + by + c = 0 first.

2. Condition Confusion:

Confusing "No solution" with "Infinite solutions".
Fix: Remember - Parallel = No solution, Coincident = Infinite

3. Word Problem Translation:

Writing "A is twice B" as x = 2 + y (wrong).
Fix: "Twice as old as" means x = 2y

9. Practice MCQs

Multiple Choice Question 1

Q1. If 4x + 6y = 12 and 2x + 3y = 6, how many solutions does the system have?
(a) Unique (b) No solution (c) Infinite (d) None

Answer: (c) Infinite (because all ratios equal 2)

Multiple Choice Question 2

Q2. Assertion: 2x + 3y = 7 and 6x + 9y = 20 has no solution.
Reason: If a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the system is inconsistent.

Answer: Both A and R are true, R is correct explanation

10. FAQ

Q: Is Chapter 3 important for CBSE Class 10 board?

A: Very important! It carries 4-7 marks and word problems appear every year.

Q: Which method is fastest in board exam?

A: Cross-multiplication method is fastest for large coefficients. For small coefficients, elimination method is best.

Q: Are these CBSE Class 10 Maths Notes based on NCERT?

A: Yes, 100%! These notes are prepared according to NCERT Textbook Chapter 3.

Q: How important is graphical method?

A: Important for 2-3 marks. Remember conditions of consistency.

Conclusion

These CBSE Class 10 Notes will help you master Chapter 3. Remember:

• Remember all three conditions (Unique, No solution, Infinite)
• Practice at least one example of each method
• Focus on word problems (Age, Number, Speed)
• Always verify your answers

Best of Luck for Board Exams 2025!
Next Chapter: Quadratic Equations

Last updated: February 17, 2025 | CBSE Board Exam Pattern 2025