Pair of Linear Equations in Two Variables – Class 10 Maths Chapter 3 Notes & NCERT PDF

Class 10MathematicsChapter 3NCERT book: Mathematics - Class 10

Notes and a simple summary of Chapter 3 of the NCERT Class 10 Mathematics book, with the official chapter PDF, flashcards, an MCQ quiz and an AI tutor for your doubts.

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Chapter 3: Pair of Linear Equations in Two Variables

Introduction

A pair of linear equations in two variables is two equations of the form a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 in the same two unknowns. Many real situations — shopping, ages, incomes, digits of numbers — lead to such pairs. This chapter solves them graphically and algebraically (by substitution and elimination).

Key Concepts & Theorems

Graphical meaning

Each linear equation is a straight line. For a pair of lines exactly one of these happens:

The lines intersect at one point → a unique solution (the pair is consistent).

The lines coincide → infinitely many solutions (dependent, and consistent).

The lines are parallel → no solution (inconsistent).

Test using ratios of coefficients

For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0:

a₁/a₂ ≠ b₁/b₂ → intersecting lines, exactly one solution.

a₁/a₂ = b₁/b₂ = c₁/c₂ → coincident lines, infinitely many solutions.

a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → parallel lines, no solution.

Methods Step by Step

Graphical method

For each equation find two or three points (make a table of x and y).

Plot both lines on the same axes.

Read the point of intersection, or conclude "coincident" / "parallel".

Substitution method

From one equation express one variable in terms of the other (e.g. y = …).

Substitute into the other equation to get an equation in one variable; solve it.

Substitute back to find the other variable.

If you get a true statement with no variable (e.g. 18 = 18) there are infinitely many solutions; a false one (e.g. 0 = 4) means no solution.

Elimination method

Multiply the equations so that the coefficients of one variable become equal (numerically).

Add or subtract to eliminate that variable.

Solve the resulting one-variable equation and substitute back.

Again, an always-true statement → infinitely many solutions; a false one → no solution.

Worked Examples

1. Champa's pants and skirts (graphical). Let x = pants, y = skirts. "Skirts are two less than twice the pants": y = 2x − 2. "Skirts are four less than four times the pants": y = 4x − 4. The lines meet at (1, 0): she bought 1 pant and no skirt.

2. Aftab and his daughter (substitution). Let s and t be their ages. Seven years ago: s − 7 = 7(t − 7), i.e. s − 7t + 42 = 0. Three years from now: s + 3 = 3(t + 3), i.e. s − 3t = 6. From the second, s = 3t + 6; substituting, 3t + 6 − 7t + 42 = 0, so t = 12, s = 42.

3. Pencils and erasers. 2x + 3y = 9 and 4x + 6y = 18. Substituting x = (9 − 3y)/2 gives 18 = 18 — true for all y. Here a₁/a₂ = b₁/b₂ = c₁/c₂ = 1/2, so the lines coincide: infinitely many solutions.

4. Incomes (elimination). Incomes ₹9x and ₹7x, expenditures ₹4y and ₹3y, each saves ₹2000: 9x − 4y = 2000 and 7x − 3y = 2000. Multiply by 3 and 4 and subtract: x = 2000, then y = 4000. Monthly incomes: ₹18,000 and ₹14,000.

5. Two-digit number. Digits x (tens) and y (units). Number + reversed number = 66 → 11(x + y) = 66 → x + y = 6. Digits differ by 2 → x − y = 2 or y − x = 2. Solving gives 42 or 24 — two such numbers.

6. Finding m. Solve 2x + 3y = 11 and 2x − 4y = −24: subtracting gives 7y = 35, y = 5, x = −2. If y = mx + 3 also holds, 5 = −2m + 3, so m = −1.

Board Exam Focus

1 mark: use the ratio test to say "consistent / inconsistent" or "unique / infinitely many / no solution".

2–3 marks: solve by substitution or elimination — show each step; state the final answer in words for word problems.

3–5 marks: frame the two equations from a situation (ages, cost, fraction, digits, rectangle perimeter) and solve; the book's garden example (half perimeter 36 m, length 4 m more than width) gives 20 m × 16 m.

Graphical questions: plot at least three points per line, label the lines and mark the solution point.

Formula Sheet

General pair: a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0

Unique: a₁/a₂ ≠ b₁/b₂

Infinitely many: a₁/a₂ = b₁/b₂ = c₁/c₂

None: a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Two-digit number with tens digit x and units y: 10x + y; reversed 10y + x

Common Mistakes

Comparing ratios with the wrong signs — write both equations in the form ax + by + c = 0 first.

Stopping after finding x and forgetting to find y (or to answer the question asked).

In digit problems, forgetting that "digits differ by 2" gives two cases.

Reading an intersection point carelessly from a rough graph — use a scale and check by substitution.

💡 Key Learning Points

✓Represent a pair of linear equations a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0 as two straight lines.
✓Interpret intersecting, coincident and parallel lines as unique, infinitely many and no solutions.
✓Use the ratios a₁/a₂, b₁/b₂, c₁/c₂ to decide consistency without solving.
✓Solve pairs by the graphical, substitution and elimination methods.
✓Translate word problems (ages, costs, incomes, digits, perimeters) into a pair of equations and solve them.

👨‍🏫 Teaching Tips

→Start with Champa's pants-and-skirts puzzle and let students guess before plotting the two lines.
→Use three coloured pairs of lines (intersecting, coincident, parallel) on one chart for the three cases.
→Build a ratio-test table on the board for several pairs and have students predict, then check graphically.
→Solve the same pair by all three methods once, so students can compare efficiency.
→Highlight the 18 = 18 and 0 = 4 outcomes in substitution as signals of infinitely many / no solutions.
→Practise framing equations from word problems in groups before solving — framing is where most marks are lost.
→Give digit problems with 'differ by 2' to show why two cases must be considered.

📋 Assessment Questions

Can the student state whether 2x + 3y = 9 and 4x + 6y = 18 have one, many or no solutions, using ratios?
Can the student solve s − 7t + 42 = 0 and s − 3t = 6 by substitution?
Can the student use elimination to find the incomes in the 9 : 7 and 4 : 3 problem?
Can the student frame and solve the two-digit number problem, finding both answers?
Can the student draw two lines accurately and read their point of intersection?

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