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I. Chapter Summary

This chapter introduces the concept of linear equations in two variables and how to solve pairs of such equations. These equations represent two straight lines in a Cartesian plane. The chapter discusses graphical and algebraic methods for solving the equations, including the substitution, elimination, and cross-multiplication methods. Students also learn about the conditions for unique, infinite, or no solution. The chapter builds essential algebraic thinking skills and problem-solving strategies used in real-life contexts.

II. Key Concepts Covered

Concept	Description
Linear Equation in Two Variables	An equation of the form $ax + by + c = 0$ , where a, b, and c are real numbers and $a,, b ne 0$ .
Graphical Representation	Each equation represents a straight line. The solution of a pair is the point of intersection.
Number of Solutions
– Unique solution: Lines intersect at one point
– Infinitely many solutions: Lines are coincident
– No solution: Lines are parallel
Algebraic Methods of Solving
1. Substitution Method
2. Elimination Method
3. Cross-Multiplication Method
Consistency of System
– Consistent: At least one solution (unique/infinite)
– Inconsistent: No solution

III. Important Questions

(A) Multiple Choice Questions (1 Mark)

1. If two lines are coincident, the system has:
   a) One solution
   b) No solution
   c) Infinite solutions ✅ (PYQ 2021)
   d) Two solutions
2. Which method is best suited when one variable has the same coefficient in both equations?
   a) Substitution
   b) Elimination ✅
   c) Graphical
   d) Cross-Multiplication
3. The pair of equations $x + 2y = 5$ and $2x+4y=10$ has:
   a) No solution
   b) Unique solution
   c) Infinite solutions ✅ (PYQ 2019)
   d) None of these
4. Which condition ensures that two linear equations have no solution?
   a) $frac{a_1}{a_2} ne frac{b_1}{b_2}$
   b) $frac{a_1}{a_2} = frac{b_1}{b_2} = frac{c_1}{c_2}$
   c) $frac{a_1}{a_2} = frac{b_1}{b_2} ne frac{c_1}{c_2}$​​ ✅
   d) None of these

(B) Short Answer Questions (2/3 Marks)

1. Solve the pair of equations using substitution:
    $begin{aligned}
   x + y &= 7 \
   x – y &= 3
   end{aligned}$
2. Check graphically whether the pair of equations $begin{aligned}
   x + y &= 3 \
   2x + 2y &= 6
   end{aligned}$
   is consistent.
3. Solve using elimination:
    $begin{aligned}
   3x + 2y &= 16 \
   2x – y &= 3
   end{aligned}$
4. Find the value of k for which the system $begin{aligned}
   kx + 3y &= k – 3 \
   12x + ky &= k
   end{aligned}$
   has infinitely many solutions.

(C) Long Answer Questions (5 Marks)

1. Solve the pair of equations by cross-multiplication method:
    $begin{aligned}
   5x – 3y &= 11 \
   2x + y &= 1
   end{aligned}$
2. Draw graphs of the equations $begin{aligned}
   x + y &= 4 \
   x – y &= 2
   end{aligned}$
   . Find the solution graphically and write the nature of the pair.
3. Two numbers are such that the sum of twice the first and thrice the second is 36, and the sum of three times the first and twice the second is 33. Find the numbers.
4. A boat goes 30 km downstream and 18 km upstream in 3 hours. It goes 21 km downstream and 12 km upstream in 2 hours. Find the speed of the boat in still water and the speed of the stream.

(D) HOTS (Higher Order Thinking Skills)

1. If one equation of a pair is $ax + by = c$, find the second equation such that the pair has:
    – a) No solution
    – b) Infinite solutions
    (Justify your answer with general values.)
2. The sum of the digits of a two-digit number is 9. If 27 is subtracted from the number, the digits interchange their places. Find the number.

IV. Key Formulas/Concepts

Formula/Concept	Description
General form	$ax + by + c = 0$
Conditions for Solutions
– Unique: $frac{a_1}{a_2} ne frac{b_1}{b_2}$
– Infinite: $frac{a_1}{a_2} = frac{b_1}{b_2} = frac{c_1}{c_2}$
– No solution: $frac{a_1}{a_2} = frac{b_1}{b_2} ne frac{c_1}{c_2}​​$
Cross-Multiplication Method
$frac{x}{(b_1c_2 – b_2c_1)} = frac{y}{(c_1a_2 – c_2a_1)} = frac{1}{(a_1b_2 – a_2b_1)}$

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	Types of Questions Typically Asked
Chapter 3: Pair of Linear Equations	6–8 Marks	1 Long Answer, 2 Short Answers, 1 MCQ

VII. Previous Year Questions (PYQs)

Year	Marks	Question
2021	1	What is the solution type of two coincident lines? (Infinite)
2020	3	Solve the pair using substitution method.
2019	1	Nature of pair: $begin{aligned} x + 2y &= 5 \ 2x + 4y &= 10 end{aligned}$? (Infinite solutions)
2018	5	Word problem involving time and speed, solved by elimination.

VIII. Real-World Application Examples

Context	Application
Banking	Solving for interest rates or time using linear conditions.
Business	Budgeting and cost-revenue analysis.
Transportation	Speed and distance problems involving time equations.
Geometry	Intersection of lines, angles, and midpoints on Cartesian planes.

IX. Student Tips & Strategies for Success

Time Management

• Practice 2 methods (e.g., substitution & cross-multiplication) daily.
• Solve at least one word problem every alternate day.

Exam Preparation

• Make a comparison chart of consistency conditions.
• Practice graphs with scale accuracy.

Stress Management

• Focus on patterns in word problems.
• Revise formulas and keep a “methods chart” handy.

X. Career Guidance & Exploration (Class-Specific)

For Classes 9–10

Streams

• Science: Physics, Engineering, Data Science
• Commerce: Business Maths, Statistics
• Arts: Logical Reasoning, Competitive Aptitude

Career Paths

• Mathematician, Economist, Engineer, Urban Planner

Competitive Exams

• NTSE, IMO, CUET Math Aptitude, RMO

XI. Important Notes

• Always refer to https://cbseacademic.nic.in and https://ncert.nic.in for official updates.
• Focus more on algebraic accuracy and diagram-based questions.
• Cross-check each word problem for its system of equations before solving.