Chapter 3: Pair Of Linear Equations In Two Variables
Master linear equations through previous year questions (2014-2026).
Complete Chapter PYQs
Access all 56 questions from Sections A to E in one comprehensive practice session.
Topic-wise Revision
CBSE Board Exam Blueprint & Mark Distribution
Chapter 3 (Pair of Linear Equations in Two Variables) consistently accounts for 4 to 6 marks in the Class 10 CBSE Board exam. Questions range from Section A 1-mark objective conditions to Section C 3-mark word problems and Section E 4-mark case studies.
| Section & Format | Marks | Core Concepts Tested | High-Frequency PYQ Pattern |
|---|---|---|---|
| Section A (MCQ) | 1 Mark | Ratio conditions for unique, infinite, or no solution. | Find value of \(k\) for parallel or coincident lines. |
| Section B (Short Answer) | 2 Marks | Direct algebraic solution: Elimination or Substitution. | Solve simultaneous equations with non-zero fractional coefficients. |
| Section C (Short Answer) | 3 Marks | Application word problems (fractions, ages, digits). | Two-digit number and reversal sum; age relations 5 years hence/ago. |
| Section D / E (Long / Case) | 4–5 Marks | Speed-time-distance, upstream/downstream, or graphical. | Find coordinates of vertices and area of triangle formed by lines with axes. |
Consistency Ratios Reference
Given equations in standard form \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\):
| Ratio Condition | Graphical Representation | Algebraic Solution | System Classification |
|---|---|---|---|
| \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\) | Intersecting lines (single intersection point) | Unique solution (exactly one pair \((x,y)\)) | Consistent |
| \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\) | Coincident lines (lines overlap everywhere) | Infinitely many solutions | Consistent (Dependent) |
| \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\) | Parallel lines (lines never meet) | No solution | Inconsistent |
Board Exam Problem-Solving Checklist:
- Always write both equations in standard form with constant terms on the same side before taking ratios \(a_1/a_2\), \(b_1/b_2\), and \(c_1/c_2\).
- In elimination, multiply equations so the LCM of the chosen variable's coefficients is matched, then add or subtract based on whether their signs are opposite or identical.
- For speed and distance word problems, check units carefully (km/h vs m/s) and verify that speed in still water \(u\) exceeds current stream speed \(v\) (\(u > v > 0\)).
- Substitute final numerical values of \(x\) and \(y\) back into both original equations to ensure zero calculation errors before moving to the next question.
Related study and practice: