source: ncert_exercises
topic: algebra
class_level: class_10
chapter: ch3
exercise: ex3.5_3.6
difficulty: advanced
Class 10 | Ch3: Pair of Linear Equations | Exercise 3.5 (Cross-Multiplication) + 3.6
Cross-Multiplication Method for a₁x+b₁y+c₁=0, a₂x+b₂y+c₂=0: x/(b₁c₂-b₂c₁) = y/(c₁a₂-c₂a₁) = 1/(a₁b₂-a₂b₁)
Q1 (Ex 3.5). Solve by cross-multiplication: (i) x+y=7, 5x+12y=7 (ii) x-3y-7=0, 3x-3y-15=0 (iii) 2x+3y+8=0, 4x+6y-8=0 (iv) x(a+b)+y(a-b)=a²-2ab-b², x+y=2a-b Answer: (i) x+y-7=0, 5x+12y-7=0. x/(1×(-7)-12×(-7))=y/((-7)×5-(-7)×1)=1/(1×12-5×1). x/(−7+84)=y/(−35+7)=1/7. x=77/7=11, y=-28/7=-4. ✅ (ii) x=4, y=-1. ✅ (iii) a₁/a₂=2/4=1/2, b₁/b₂=3/6=1/2, c₁/c₂=8/-8≠1/2. Parallel lines — no solution. ✅ (iv) Solve to get x=a, y=b-a. Actually: x+y=2a-b and x(a+b)+y(a-b)=a²-2ab-b². x=a, y=a-b. ✅
Q2 (Ex 3.5). Solve: 2/x+3/y=13, 5/x-4/y=-2 (where x,y≠0). Answer: Let 1/x=p, 1/y=q. 2p+3q=13, 5p-4q=-2. Eliminate: 8p+12q=52 and 15p-12q=-6. 23p=46 → p=2, q=3. x=1/2, y=1/3. ✅
Q3 (Ex 3.6). Equations reducible to linear pair. Solve: 1/(3x+y)+1/(3x-y)=3/4, 1/(2(3x+y))-1/(2(3x-y))=-1/8. Answer: Let u=1/(3x+y), v=1/(3x-y). u+v=3/4, u/2-v/2=-1/8 → u-v=-1/4. Adding: 2u=1/2 → u=1/4, v=1/2. 3x+y=4, 3x-y=2. x=1, y=1. ✅