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CBSE · Class 10 · Mathematics · Chapter 3

Pair of Linear Equations in Two Variables — Class 10 Mathematics (CBSE) explained

  • Board exam 2026-27
  • 4 lessons
  • 28 scenes
  • About 24 min
  • Updated

Quick answer

Pair of Linear Equations in Two Variables is Chapter 3 of CBSE Class 10 Mathematics. It covers algebraic condition for coincidence, algebraic condition for parallel lines, completing substitution, solving an age model, dependent pair through substitution and inconsistent pair through substitution. AI Lessons teaches it in 4 narrated lessons (about 24 minutes) with an AI tutor and board-pattern practice.

What you’ll learn

  1. Lesson 1: Two Equations, Two Lines, Three Possibilities

    6 min
    • Turn Akhila's Fair Visit into Two Equations
    • Read the Number of Solutions from the Lines
    • Unequal Variable-Coefficient Ratios Mean Intersection
    • Three Equal Ratios Mean Coincident Lines
    • A Different Constant Ratio Means Parallel Lines
    • Plot Two Lines and Read Their Common Point
    • Scaling Reveals the Same Line

  2. Lesson 2: From Graphs to Substitution

    7 min
    • Interpret Champa's Shopping Conditions
    • Choose the Right Graphical Practice Strategy
    • Graphs Are Not Always Convenient for Exact Coordinates
    • Start Substitution with the Convenient Equation
    • Substitute, Solve and Recover the Other Variable
    • Use the Three-Step Substitution Method
    • Shift Both Ages to the Same Time
    • Find the Present Ages by Substitution

  3. Lesson 3: What Substitution Reveals and How Elimination Works

    6 min
    • A True Statement Does Not Determine Unique Prices
    • A False Statement Means the Rails Do Not Cross
    • Keep the Substitution Method Steady Across Different Forms
    • Use Ratios to Represent Income and Expenditure
    • Equalise Coefficients and Eliminate y
    • Recover y and Verify the Financial Conditions
    • Choose Multipliers and Read the Elimination Outcome

  4. Lesson 4: Elimination, Situational Problems and Practice

    5 min
    • Elimination Can Prove That No Solution Exists
    • Place Value Turns Reversed Digits into Equations
    • Two Digit Orders Give Two Numbers
    • Compare Two Algebraic Routes for the Same Pair
    • Identify Counts, Charges and Expressions Before Solving
    • Recap: Model, Classify, Solve and Verify

Key concepts

Algebraic condition for coincidence
The equations are equivalent, so their lines coincide. Every point on that shared line satisfies both equations.
Algebraic condition for parallel lines
The lines are parallel, so there is no common solution.
Completing substitution
An exact fractional answer is still a valid common solution.
Solving an age model
Aftab is forty two years old, and his daughter is twelve years old.
Dependent pair through substitution
The equations are equivalent and have infinitely many solutions.
Inconsistent pair through substitution
The pair has no common solution and is inconsistent.
Place-value modelling
The number and its reversal have a sum of sixty six.
Final situational applications
Define the unknowns carefully, form the two conditions, and interpret any solution in the original situation.

Exam focus

Board exam
CBSE assesses this chapter in the 2026-27 year-end board examination.
Syllabus
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES: 1. Pair of linear equations in two variables and graphical method of their solution, consistency/inconsistency. 2. Algebraic conditions for number of solutions. 3. Solution of a pair of linear equations in two variables algebraically - by substitution, by elimination. Simple situational problems. Competencies: describes plotting a pair of linear equations and graphically finding the solution; models and solves contextualised problems using equations (e.g., simultaneous linear equations in two variables). Learning outcome: find the solution of pair of linear equations in two variables graphically and algebraically (substitution and elimination method).
Exam note
Algebraic methods in the syllabus are substitution and elimination only; the 2026-27 textbook (3.3) also teaches only these two.
Exam note
Situational problems should stay "simple", as in the syllabus wording.
Paper pattern
Mathematics Standard: 80 marks, 3 hours, 38 questions. Section A: 18 MCQs + 2 assertion-reason (1 mark each); B: 5 VSA (2 marks); C: 6 SA (3 marks); D: 4 LA (5 marks); E: 3 case-study questions (4 marks). No calculators.
Question style
About 50% of board questions are competency-focused.
Practice in the app
94 board-pattern questions with marking schemes: 23 multiple choice (1 mark), 13 assertion–reason (1 mark), 21 very short answer (2 marks), 22 short answer (3 marks), 5 long answer (5 marks) and 10 case-based (4 marks).

Sample questions with marking points

Board-pattern questions from the AI Lessons practice set for this chapter. The marking points show what an examiner looks for; full model answers are in the app.

  1. Very short answer2 marks

    A student proposes x = 49/29 and y = 19/29 for 7x - 15y = 2 and x + 2y = 3. Verify the proposal using both equations.

    Marking points

    • Correctly substitute into and verify 7x - 15y = 2.1
    • Correctly substitute into and verify x + 2y = 3.1

  2. Short answer3 marks

    A student solves 2x + 3y = 11 and 2x - 4y = -24, then claims that their common solution satisfies y = mx + 3 when m = 1. Check the claim by finding the common solution and the correct value of m.

    Marking points

    • Subtract the equations correctly to obtain 7y = 35.1
    • Obtain y = 5.0.5
    • Substitute back to obtain x = -2.0.5
    • Substitute the common solution into y = mx + 3 to obtain 5 = -2m + 3.0.5
    • Obtain m = -1 and reject the claim m = 1.0.5

  3. Long answer5 marks

    Two people's monthly incomes are in the ratio 9 : 7 and their expenditures are in the ratio 4 : 3. Each saves ₹ 2000 per month. A student reports ₹ 2000 and ₹ 4000 as their monthly incomes. Form suitable equations and solve by elimination to correct the report. Verify the actual incomes against both given ratios.

    Marking points

    • Represent incomes by 9x and 7x, and expenditures by 4y and 3y.0.5
    • Form 9x - 4y = 2000.0.5
    • Form 7x - 3y = 2000.0.5
    • Scale the first equation correctly to 27x - 12y = 6000.0.5
    • Scale the second equation correctly to 28x - 12y = 8000.0.5
    • Subtract correctly to obtain x = 2000.0.5
    • Back-substitute into an original equation and obtain y = 4000.0.5
    • Correct the report by giving actual incomes of ₹ 18000 and ₹ 14000 rather than the auxiliary values x and y.0.5
    • Verify the income ratio 18000 : 14000 = 9 : 7.0.5
    • Use the savings to obtain expenditures ₹ 16000 and ₹ 12000 and verify their ratio 4 : 3.0.5

Frequently asked questions

What does Pair of Linear Equations in Two Variables cover in Class 10 Mathematics?

It covers algebraic condition for coincidence, algebraic condition for parallel lines, completing substitution, solving an age model, dependent pair through substitution and inconsistent pair through substitution. AI Lessons teaches it in 4 lessons: Two Equations, Two Lines, Three Possibilities, From Graphs to Substitution, What Substitution Reveals and How Elimination Works and Elimination, Situational Problems and Practice.

Is Pair of Linear Equations in Two Variables in the CBSE Class 10 board exam?

Yes. CBSE assesses Pair of Linear Equations in Two Variables in the 2026-27 year-end board examination.

For the pair 9x + 3y + 12 = 0 and 18x + 6y + 24 = 0, which conclusion follows from comparing the corresponding coefficient ratios?

All three ratios equal 1/2, so there are infinitely many solutions. The ratios 9/18, 3/6 and 12/24 all equal 1/2, so the lines coincide and have infinitely many common solutions.

For 7x - 15y = 2 and x + 2y = 3, which expression follows from the second equation and is ready to replace x in the first?

X = 3 - 2y. Subtracting 2y from both sides of x + 2y = 3 gives x = 3 - 2y.

What does false statement mean?

A false statement means the pair is inconsistent.

For the pair 0.2x + 0.3y = 1.3 and 0.4x + 0.5y = 2.3, which expression for x follows from the first equation?

X = (1.3 - 0.3y)/0.2. Subtracting 0.3y gives 0.2x = 1.3 - 0.3y, so the entire right-hand side must be divided by 0.2.

How does AI Lessons teach Pair of Linear Equations in Two Variables?

As 4 narrated, animated lessons (28 scenes, about 24 minutes in all), with an AI tutor that knows the step you are on and board-pattern practice with marking schemes. The full lessons open after sign-in on an AI Lessons account.

Watch the full lesson

Pair of Linear Equations in Two Variables, as a narrated, animated lesson

4 lessons · 28 animated scenes · about 24 minutes, with an AI tutor that knows the step you are on and 94 board-pattern questions with marking schemes.

Lessons open after sign-in. A parent or guardian holds the account and adds each child.

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Last updated . Written by the AI Lessons team from our own lesson scripts and practice questions, aligned to the Central Board of Secondary Education syllabus for 2026-27. Always check your school’s latest syllabus.