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Class 9 → Mathematics → Chapter 4: Linear Equations in Two Variables
I. Chapter Summary
The chapter Linear Equations in Two Variables introduces equations involving two variables (usually x and y) of the form ax + by + c = 0, where a, b, and c are constants. Students learn how to find solutions, represent them graphically, and understand that such equations represent straight lines. The chapter emphasizes real-life applications like cost calculations, distance-time relationships, and budgeting.
II. Key Concepts Covered
1. Linear Equation in Two Variables
An equation of the form:
ax + by + c = 0, where a and b are not both zero.
2. Solution of Linear Equation
A pair (x, y) that satisfies the equation.
Example:
For x + y = 5 → (2,3), (1,4) are solutions.
3. Infinite Solutions
A linear equation has infinitely many solutions.
4. Graph of Linear Equation
- Graph is a straight line
- Each point on line is a solution
5. Plotting Graph
Steps:
- Find at least two solutions
- Plot them on graph
- Join points to form line
6. Special Cases
- x = a → vertical line
- y = b → horizontal line
III. Important Questions
(A) MCQs (1 Mark)
- The general form of linear equation in two variables is:
(a) ax² + by² = 0
(b) ax + by + c = 0
(c) ax³ + by = 0
(d) x² + y = 0
Answer: (b) - The graph of linear equation is:
(a) Circle (b) Parabola (c) Straight line (d) Curve
Answer: (c) - Number of solutions of linear equation in two variables is:
(a) 1 (b) 2 (c) Infinite (d) Zero
Answer: (c) - Equation x = 3 represents:
(a) Horizontal line (b) Vertical line (c) Curve (d) Circle
Answer: (b)
(B) Short Answer Questions (2/3 Marks)
- Find three solutions of equation x + 2y = 6. (CBSE 2020)
- Check whether (3,2) is a solution of 2x + y = 8.
- Write equation of line parallel to x-axis.
- Plot two points satisfying 3x + y = 6.
(C) Long Answer Questions (5 Marks)
- Draw graph of equation x + y = 5 and verify solutions. (CBSE 2019)
- Find solutions of 2x – y = 4 and represent graphically.
- Explain graphical method of solving linear equations.
- Plot graph of x = 2 and y = 3 and interpret.
(D) HOTS Questions
- Can a linear equation represent a single point? Justify your answer.
- If two linear equations represent the same line, what can you conclude about their solutions?
IV. Key Formulas / Concepts
- General form:
ax + by + c = 0 - Solution:
Any (x, y) satisfying equation - Graph:
- Straight line
- Infinite solutions
- Special equations:
- x = a → vertical line
- y = b → horizontal line
Example:
For x + y = 4
Solutions: (0,4), (1,3), (2,2)
V. Deleted Portions (CBSE 2025–2026)
No portions have been deleted from this chapter as per the rationalized NCERT textbooks.
VI. Chapter-Wise Marks Bifurcation (Estimated – CBSE 2025–2026)
| Unit/Chapter | Estimated Marks | Type of Questions Typically Asked |
|---|---|---|
| Linear Equations in Two Variables | 6–8 marks | MCQs, graph-based questions, short answer |
Note: This is an estimate. Actual marks may vary.
VII. Previous Year Questions (PYQs)
1 Mark
- Identify type of equation (CBSE 2021)
2/3 Marks
- Find solutions of equation (CBSE 2020)
- Verify ordered pair (CBSE 2019)
5 Marks
- Draw graph of linear equation (CBSE 2018)
- Represent equations graphically (CBSE 2022)
VIII. Real-World Application Examples
- Used in budget planning (income vs expenses)
- Helps in distance-time problems
- Applied in business profit calculations
- Used in physics motion graphs
IX. Student Tips & Strategies for Success
Time Management
- Practice graph questions regularly
- Revise formulas daily
Exam Preparation
- Draw neat graphs with scale
- Label axes properly
Stress Management
- Practice step-by-step solutions
- Avoid calculation errors
X. Career Guidance & Exploration (Class 9 Level)
Streams After Class 10:
- Science → Engineering, Data Science
- Commerce → Business, Economics
- Arts → Statistics, Geography
Entrance Exams (Basic Awareness):
- JEE
- CUET
- NDA
XI. Important Notes
- Always solve using stepwise method
- Practice graph drawing carefully
- Focus on concept, not memorization
- Revise NCERT examples regularly
- Check CBSE updates from official website