Class 9 Mathematics Chapter 3: Pair of Linear Equations in Two Variables — Previous Year Questions (2020–2025)

Reading Chapter 3 notes one more time won't change your exam score—but solving 5 previous papers will. Here's why: (1) **Pattern recognition**: Examiners reuse question structures. You'll see 'the consistent system graphed as intersecting lines' appear in 3 different years, but phrased differently each time. Solve 3 versions, and you'll spot it instantly on exam day. (2) **Method selection speed**: Should you use graphical or substitution here? Theory doesn't tell you—but doing 10 questions does. Your brain learns which method feels fastest for different equation types. (3) **Marking scheme internals**: A 3-mark question about 'proving consistency' always expects: statement of condition (1 mark) + graphical or algebraic proof (1.5 marks) + conclusion (0.5 marks). Reading the mark scheme teaches you faster than re-reading textbook derivations. (4) **Confidence in your weakness**: If you've solved elimination problems in 3 papers and still fumble, you know exactly what to drill—not guess. This focused practice is what moves you from 60 to 85 in 2 weeks.

**Question 1:** For the pair of equations 2x + 3y = 5 and 4x + 6y = 10, the system is **(A) consistent and dependent (B) inconsistent (C) consistent and independent (D) no solution**. **Solution:** Divide the second equation by 2: 2x + 3y = 5. Both equations are identical, so every point on the line 2x + 3y = 5 is a solution. The system is **consistent and dependent** (infinitely many solutions). ✓ **Why it repeats**: Tests whether students confuse 'dependent' with 'no solution.' **Question 2:** The graphical representation of y = 2x + 1 and y = 2x − 3 gives **(A) intersecting lines (B) parallel lines (C) coincident lines (D) perpendicular lines**. **Solution:** Both lines have the same slope (2) but different y-intercepts (1 ≠ −3). **Parallel lines** (no solution, inconsistent). ✓ **Question 3:** If 3x + 4y = 12 and x − 2y = −2 intersect at point P, then P lies in which quadrant? **Solution:** Solve: From equation 2, x = 2y − 2. Substitute: 3(2y − 2) + 4y = 12 → 10y = 18 → y = 1.8, x = 1.6. Point (1.6, 1.8) is in **Quadrant I**. ✓ **Question 4:** The equations x + 2y = 6 and 2x + 4y = 12 represent **(A) two different lines (B) the same line (C) perpendicular lines (D) intersecting at origin**. **Solution:** The second equation is 2 × (first equation). **Same line** (coincident), consistent and dependent. ✓ **Question 5:** For ax + by = c and dx + ey = f, if a/d = b/e ≠ c/f, the system is **(A) consistent (B) inconsistent (C) dependent (D) unique**. **Solution:** Parallel lines (same slope, different intercepts). **Inconsistent** (no solution). ✓

**Question 1 – Substitution Method:** Solve the pair 2x + y = 7 and x − y = 2 using substitution. **Solution:** From equation 2: x = y + 2. Substitute into equation 1: 2(y + 2) + y = 7 → 3y + 4 = 7 → y = 1. Then x = 1 + 2 = 3. **Verification:** 2(3) + 1 = 7 ✓ and 3 − 1 = 2 ✓. **Answer:** x = 3, y = 1. ✓ (Typically: 0.5 marks for substitution setup, 1 mark for solving y, 1 mark for x, 0.5 for verification.) **Question 2 – Elimination Method:** Solve 3x + 2y = 11 and 2x − 3y = 3 using elimination. **Solution:** Multiply equation 1 by 3: 9x + 6y = 33. Multiply equation 2 by 2: 4x − 6y = 6. Add: 13x = 39 → x = 3. Substitute: 2(3) − 3y = 3 → y = 1. **Answer:** x = 3, y = 1. ✓ (1 mark for choosing multipliers, 1 mark for elimination step, 1 mark for final answer.) **Question 3 – Graphical Representation & Consistency:** Plot the lines x + y = 5 and x − y = 1. Are they consistent? Find the solution. **Solution:** Line 1 passes through (0, 5) and (5, 0). Line 2 passes through (0, −1) and (1, 0). They intersect at (3, 2). Solve: Add equations → 2x = 6 → x = 3, y = 2. System is **consistent and independent** (unique solution). ✓ (1 mark for correct graphs, 1 mark for identifying intersection, 1 mark for algebraic solution.) **Question 4 – Cross-Multiplication Method:** Solve using cross-multiplication: 5x + 8y = 9 and 2x + 3y = 4. **Solution:** Using a₁x + b₁y = c₁ and a₂x + b₂y = c₂, cross-multiply: $$\frac{x}{b_1 c_2 - b_2 c_1} = \frac{y}{c_1 a_2 - c_2 a_1} = \frac{1}{a_1 b_2 - a_2 b_1}$$ With a₁ = 5, b₁ = 8, c₁ = 9, a₂ = 2, b₂ = 3, c₂ = 4: $$\frac{x}{8(4) - 3(9)} = \frac{y}{9(2) - 4(5)} = \frac{1}{5(3) - 2(8)}$$ $$\frac{x}{32 - 27} = \frac{y}{18 - 20} = \frac{1}{15 - 16}$$ $$\frac{x}{5} = \frac{y}{-2} = \frac{1}{-1}$$ x = −5, y = 2. ✓ (Conceptual depth tested here.) **Question 5 – Real-World Application:** Two numbers sum to 15, and their difference is 3. Form equations and solve. **Solution:** Let the numbers be x and y. x + y = 15 and x − y = 3. Add: 2x = 18 → x = 9. Then y = 6. **Answer:** The numbers are 9 and 6. ✓ (0.5 marks equation formation, 1.5 marks solving, 1 mark interpretation.)

**Question 1 – Solve and Verify by Graphical & Algebraic Method:** The lines 2x − y − 2 = 0 and x + y − 4 = 0 intersect at point P. Find P algebraically. Then prove that P lies on both lines graphically. **Full Solution:** **Algebraic:** From equation 1: y = 2x − 2. Substitute into equation 2: x + (2x − 2) − 4 = 0 → 3x − 6 = 0 → x = 2. Then y = 2(2) − 2 = 2. **Point P = (2, 2).** ✓ **(1 mark for setup, 1.5 marks for correct solving)** **Verification:** 2(2) − 2 − 2 = 0 ✓ and 2 + 2 − 4 = 0 ✓ **(0.5 marks)** **Graphical:** Line 1 (y = 2x − 2) passes through (0, −2) and (1, 0). Line 2 (y = −x + 4) passes through (0, 4) and (4, 0). Both lines intersect at (2, 2) on the graph. System is **consistent and independent.** **(2 marks for correct graphs and intersection point identification)** **Total: 5 marks.** ✓ **Question 2 – Word Problem with Two Variables:** A boat travels 30 km downstream in 2 hours and 20 km upstream in the same time. Find the speed of the boat in still water and the speed of the current. **Full Solution:** Let b = speed of boat in still water, c = speed of current. Downstream: (b + c) = 30/2 = 15 km/h. Upstream: (b − c) = 20/2 = 10 km/h. **Equations:** b + c = 15 and b − c = 10. **Solve by addition:** 2b = 25 → b = 12.5 km/h. Then c = 2.5 km/h. **(1.5 marks for equation formation, 2 marks for solving method, 1 mark for final answer and units, 0.5 marks for verification: downstream 15 km/h ✓, upstream 10 km/h ✓)** **Total: 5 marks.** ✓ **Question 3 – Consistency & Inconsistency Analysis:** For what value of k is the system 2x + ky = 5 and 3x + (k + 2)y = 7 (a) consistent and independent, (b) inconsistent? **Full Solution:** (a) **Consistent and independent:** a₁/a₂ ≠ b₁/b₂ → 2/3 ≠ k/(k + 2). Cross-multiply: 2(k + 2) ≠ 3k → 2k + 4 ≠ 3k → **k ≠ 4.** For **any k ≠ 4**, the system is consistent and independent. **(1.5 marks)** (b) **Inconsistent:** Parallel lines → a₁/a₂ = b₁/b₂ ≠ c₁/c₂. We need 2/3 = k/(k + 2) and 2/3 ≠ 5/7. From 2/3 = k/(k + 2) → 2(k + 2) = 3k → **k = 4.** Check: When k = 4, is 2/3 ≠ 5/7? Yes (2/3 ≈ 0.67, 5/7 ≈ 0.71). **Inconsistent when k = 4.** **(3 marks for reasoning and verification)** **Total: 5 marks.** ✓

The rationalized CBSE Class 9 Mathematics syllabus (2024–25 onwards) has introduced subtle shifts in how Chapter 3 is examined: **(1) Emphasis on graphical intuition over heavy algebra:** Newer papers spend more 1-mark and 3-mark questions asking students to *identify consistency at a glance* (e.g., 'Are these parallel, intersecting, or coincident?') rather than solve fully. This rewards students who sketch graphs mentally—not just calculate. **(2) Real-world applications dominate 5-mark slots:** Word problems (speed-distance, cost-quantity, age-relation) now comprise ~60% of 5-mark sections. Pure algebraic systems appear less often. **(3) Cross-multiplication as a 'bonus method,' not core:** While still in the syllabus, cross-multiplication (determinant-based) has been de-emphasized in recent papers. Substitution and elimination remain the primary expected methods. **(4) Consistency language precision:** Examiners now mark strictly on terminology: 'Dependent' doesn't mean 'has many solutions'—it means 'dependent system (infinitely many solutions and consistent).' Confusion here costs marks. **(5) Proof-based 3-mark questions rising:** Questions like 'Prove that these equations represent intersecting lines' (without numerical solving) are trending upward. This tests conceptual understanding over computational speed. To align your prep with 2026–27 patterns, balance your question mix: 40% graphical reasoning, 30% standard solving, 30% word problems.

**Read the question carefully (10 seconds):** Identify what's being asked: solve? Prove consistency? Graph? Interpret? Don't assume. **1-mark questions (under 1 minute each):** (a) If it asks for consistency, mentally check slopes: same slope + same intercept = coincident (dependent). Same slope + different intercept = parallel (inconsistent). Different slopes = intersecting (unique solution). No calculation needed. (b) If it's a definition ('What does inconsistent mean?'), recall: **Inconsistent = no solution = parallel lines = a₁/a₂ = b₁/b₂ ≠ c₁/c₂.** **3-mark questions (3–4 minutes each):** (a) If it says 'solve by [method],' don't use another method—examiners mark method strictly. (b) For substitution: choose the simpler variable from the simpler equation. (c) For elimination: find LCM of coefficients to cancel, not random multiples. (d) Always verify your answer by substituting back into *both* original equations. (e) **Common mistake:** Forgetting to simplify or writing x = 3, y = 4 without checking 2(3) + 5(4) = 26 (does it match?). **5-mark questions (6–7 minutes each):** (a) **Word problems:** Define variables *clearly* (e.g., 'Let x = cost of 1 pen, y = cost of 1 pencil'). Write equations. Solve. Interpret the answer in context ('Cost of 1 pen = Rs 5'). Missing this costs 1 mark. (b) **Graphical + algebraic proof:** Always do both. Graphical alone = incomplete. (c) **Consistency proof:** State the condition (a₁/a₂ = b₁/b₂ ≠ c₁/c₂ for inconsistent, etc.), substitute values, conclude. This earns full marks. (d) **Sketch time:** If a graph is requested, don't spend 2 minutes perfecting axes—rough lines with labeled intercepts are fine. **Under time pressure:** Attempt all 1-mark and 3-mark questions first. 5-mark word problems last (they're time-heavy). **Start a 3-day free trial at cbsetutor.ai** to access interactive simulations of graphing and step-by-step feedback on your solving method—the fastest way to build exam-day reflexes.