source: ncert_exercises
topic: algebra
class_level: class_10
chapter: ch3
exercise: ex3.2
difficulty: intermediate
Class 10 | Ch3: Pair of Linear Equations | Exercise 3.2 (Graphical Method)
Q1. Solve graphically: (i) x-y+1=0, 3x+2y-12=0 (ii) 2x+y-6=0, 4x-2y-4=0 (iii) x-y-1=0, 2x+y-8=0 Answer: (i) Find points: x-y+1=0 → (0,1),(1,2),(2,3). 3x+2y=12 → (0,6),(2,3),(4,0). Intersection: x=2,y=3. ✅ (ii) 2x+y=6: (0,6),(3,0). 4x-2y=4 → 2x-y=2: (0,-2),(1,0). Intersection: x=2,y=2. ✅ (iii) x-y=1: (0,-1),(1,0). 2x+y=8: (0,8),(4,0). Intersection: x=3,y=2. ✅
Q2. Graphically find whether consistent/inconsistent: (i) x+y=5; 2x+2y=10 (ii) x-y=8; 3x-3y=16 (iii) 2x+y-6=0; 4x-2y-4=0 (iv) 2x-2y-2=0; 4x-4y-5=0 Answer: (i) a₁/a₂=1/2, b₁/b₂=1/2, c₁/c₂=5/10=1/2. All equal → coincident → infinite solutions → consistent. ✅ (ii) a₁/a₂=1/3, b₁/b₂=-1/-3=1/3, c₁/c₂=-8/-16=1/2. a/a=b/b≠c/c → parallel → inconsistent. ✅ (iii) a₁/a₂=2/4=1/2, b₁/b₂=1/-2≠1/2 → intersecting → consistent, unique solution. ✅ (iv) a₁/a₂=2/4=1/2, b₁/b₂=-2/-4=1/2, c₁/c₂=-2/-5=2/5. a/a=b/b≠c/c → parallel → inconsistent. ✅
Q3. The cost of 2 pencils and 3 erasers = ₹9, 4 pencils and 6 erasers = ₹18. Find if consistent. Answer: 2x+3y=9 and 4x+6y=18. a₁/a₂=2/4=1/2, b₁/b₂=3/6=1/2, c₁/c₂=9/18=1/2. All equal → coincident → infinite solutions. Cannot find unique cost. ✅