NCERT Class 10 Mathematics Mathematics: Chapter 3 — PAIR OF LINEAR EQUATIONS IN TWO VARIABLES
This chapter introduces the concept of Pair of Linear Equations in Two Variables, building upon knowledge from Class IX. It presents a real-world scenario involving Akhila at a fair to illustrate how such problems can be represented by two linear equations. The chapter defines consistent and inconsistent pairs of linear equations, and dependent pairs. It then details the graphical method for solving these equations, explaining the three possible outcomes: intersecting lines with a unique solution, parallel lines with no solution, and coincident lines with infinitely many solutions. Examples are provided to demonstrate each case, comparing the coefficients of the equations to determine their nature and solution set, aiding students in understanding algebraic problem-solving.
Quick info
| Board | CBSE / NCERT |
|---|---|
| Class | Class 10 |
| Subject | Mathematics |
| Book | Mathematics |
| Chapter | Chapter 3 — PAIR OF LINEAR EQUATIONS IN TWO VARIABLES |
| Language | English |
| PDF type | NCERT Textbook |
| Session | CBSE 2026 |
| Reading time | 3 minutes |
| Word count | 539 |
Learning outcomes
- Represent real-world problems using pairs of linear equations in two variables.
- Understand the concepts of consistent, inconsistent, and dependent pairs of linear equations.
- Solve pairs of linear equations graphically.
- Determine the nature of solutions (unique, no solution, infinitely many) based on graphical representations.
- Compare coefficients to analyze the behavior of lines and the existence of solutions.
Vocabulary
| Word | Meaning |
|---|---|
| Linear equation in two variables | An equation of the form ax + by + c = 0, where a, b, and c are real numbers and at least one of a or b is non-zero. |
| Pair of linear equations in two variables | A system of two linear equations in two variables, typically represented as a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0. |
| Consistent pair of linear equations | A pair of linear equations that has at least one solution. |
| Inconsistent pair of linear equations | A pair of linear equations that has no solution. |
| Dependent pair of linear equations | A pair of linear equations that has infinitely many distinct common solutions; always consistent. |
| Unique solution | A single ordered pair (x, y) that satisfies both equations in a system. |
| Infinitely many solutions | An unlimited number of ordered pairs (x, y) that satisfy both equations in a system. |
| Coincident lines | Lines that lie exactly on top of each other, representing infinitely many solutions. |
| Parallel lines | Lines that never intersect, representing no solution. |
| Intersecting lines | Lines that cross each other at a single point, representing a unique solution. |
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Practice questions
- Represent the following situation as a pair of linear equations: Ritu can row downstream 20 km in 2 hours and upstream 4 km in 2 hours. Find her speed of rowing in still water and the speed of the current. Answer: Let Ritu's speed in still water be x km/hr and the speed of the current be y km/hr. Downstream speed = (x+y) km/hr. Upstream speed = (x-y) km/hr. Equations: 2(x+y) = 20 and 2(x-y) = 4.
- Determine if the pair of equations x + 2y = 4 and 2x + 4y = 12 has a unique solution, no solution, or infinitely many solutions. Answer: No solution, as the lines are parallel (a1/a2 = b1/b2 ≠ c1/c2).
- What is a consistent pair of linear equations? Answer: A pair of linear equations that has at least one solution.
- If the lines representing two linear equations are coincident, how many solutions does the pair have? Answer: Infinitely many solutions.
Frequently asked questions
What is a pair of linear equations in two variables?
It is a system of two linear equations, each involving two distinct variables, typically represented in the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0.
What does it mean for a pair of linear equations to be consistent?
A pair of linear equations is consistent if it has at least one solution. This occurs when the lines representing the equations intersect at a single point or are coincident.
When do two linear equations have no solution?
Two linear equations have no solution when the lines representing them are parallel and distinct. This happens when the ratio of coefficients a1/a2 = b1/b2 but a1/a2 ≠ c1/c2.
What is a dependent pair of linear equations?
A dependent pair of linear equations is one that has infinitely many solutions. This occurs when the lines representing the equations are coincident (the same line).
How can we determine the nature of solutions graphically?
Graphically, if the lines intersect at one point, there's a unique solution. If the lines are parallel, there's no solution. If the lines are coincident, there are infinitely many solutions.
What is the condition for lines to be coincident?
Lines are coincident if the ratios of their corresponding coefficients are equal: a1/a2 = b1/b2 = c1/c2.
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NCERT Class 10 Mathematics — Mathematics — Chapter 3 — PAIR OF LINEAR EQUATIONS IN TWO VARIABLES. Verified by NCERT Help Editorial Team. Reviewed on 29 Jul 2026. Last updated 10 Aug 2026.