Revision Notes: Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables

A fast, one-glance recap of Class 10 Maths Chapter 3 (Pair of Linear Equations in Two Variables) — for the full worked explanations, see the Solutions. These Class 10 Mathematics Chapter 3 notes are ideal for quick revision just before exams.

Last Updated: October 24, 2026

Common Mistakes Students Make in Pair of Linear Equations in Two Variables

Why This Chapter Matters

Pair of Linear Equations in Two Variables is one of the standard chapters in the CBSE/NCERT Class 10 Maths syllabus and questions from it appear regularly in board exams, in formats ranging from 1-mark objective questions to longer 3-5 mark descriptive or numerical questions. Because the concepts here often build on earlier chapters (and feed into later ones), a solid grip on this chapter also makes the rest of the Maths syllabus easier to follow.

Revision Notes: Pair of Linear Equations in Two Variables

• Linear equation in two variables: ax + by + c = 0 (a, b not both zero)
• Pair of linear equations: a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
• Graphical meaning: two lines can be intersecting, parallel, or coincident
• Ratio test: a₁/a₂ ≠ b₁/b₂ → intersecting, consistent, unique solution
• a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → parallel, inconsistent, no solution
• a₁/a₂ = b₁/b₂ = c₁/c₂ → coincident, consistent, infinitely many solutions
• Algebraic methods: substitution (express one variable in terms of the other and substitute) and elimination (make one variable’s coefficient equal in both equations, then add/subtract)
• Word problems: define both variables clearly first, form two equations from the given conditions, solve, then sanity-check the answer fits the context (ages, speeds, costs must be positive)
• Reciprocal substitution trick: for speed/time problems with terms like 1/(x−y) and 1/(x+y), substitute u = 1/(x−y), v = 1/(x+y) to reduce to a simple linear pair
• Know by heart: the ratio-test table above is a near-guaranteed 1-2 mark question every year

See also: Extra Questions (HOTS) | Formulas Handbook

How to Use These Notes

These revision notes are meant for quick recall in the last few days before an exam or unit test — they are not a substitute for first reading the full chapter and working through examples in the NCERT textbook. A good routine is: read the chapter once, solve the in-text and end-of-chapter questions using the linked Solutions, then come back to this page a day or two before the exam to refresh the key definitions and formulas without re-reading the whole chapter.

Quick visual: a worked diagram from the full Solutions page, for reference.

Frequently Asked Questions

What are the three possible outcomes when solving a pair of linear equations graphically?
The lines can intersect at exactly one point (a unique solution, consistent), be parallel and never meet (no solution, inconsistent), or coincide completely (infinitely many solutions, dependent and consistent).

How can you tell which of these three cases applies without graphing, just from the equations?
By comparing the ratios of coefficients (a1/a2, b1/b2, c1/c2): if a1/a2 does not equal b1/b2, there is a unique solution; if a1/a2 = b1/b2 but not equal to c1/c2, there is no solution; if all three ratios are equal, there are infinitely many solutions.

What to Revise First (and Last) in This Chapter

Written by Satish

NCERTBooks.org is an independent educational resource run by a small team focused on making official NCERT textbooks easy to find, read, and download for students, parents, and teachers across India. We are not affiliated with NCERT or the Ministry of Education -- we organise publicly available NCERT content by class and subject, verify links against official sources, and build tools (like our in-browser reader) that make studying more convenient. Every guide we publish is written and reviewed by our team based on the actual NCERT curriculum and syllabus.