source: ncert
topic: algebra
class_level: class_10
chapter: ch3
difficulty: intermediate
Class 10 | Ch3: Pair of Linear Equations in Two Variables STANDARD FORM: a1x+b1y+c1=0 and a2x+b2y+c2=0. CONSISTENCY CONDITIONS: Unique solution (consistent): a1/a2 ≠ b1/b2. Lines intersect. Infinitely many solutions (consistent, dependent): a1/a2=b1/b2=c1/c2. Lines coincide. No solution (inconsistent): a1/a2=b1/b2 ≠ c1/c2. Lines parallel. METHODS OF SOLVING: Graphical: plot both lines, find intersection point. Substitution: express one variable in terms of other, substitute. Elimination: multiply equations to make coefficients equal, add/subtract. Cross-multiplication: x/((b1c2-b2c1))=y/((c1a2-c2a1))=1/((a1b2-a2b1)). CROSS MULTIPLICATION FORMULA: For a1x+b1y+c1=0 and a2x+b2y+c2=0: x=(b1c2-b2c1)/(a1b2-a2b1), y=(c1a2-c2a1)/(a1b2-a2b1). SOLVED EXAMPLES: Example 1: Solve 2x+3y=11 and 2x-4y=-24 by elimination. Subtract: 7y=35 → y=5. Substitute: 2x+15=11 → x=-2. Solution: x=-2, y=5. Example 2: Solve by cross-multiplication: 2x+3y-11=0 and 2x-4y+24=0. x/(3×24-(-4)×(-11))=y/((-11)×2-24×2)=1/(2×(-4)-2×3). x/(72-44)=y/(-22-48)=1/(-8-6). x/28=y/(-70)=1/(-14). x=28/(-14)=-2. y=-70/(-14)=5. Solution: (-2,5). Example 3: Check 3x+2y=5 and 6x+4y=10. a1/a2=3/6=1/2. b1/b2=2/4=1/2. c1/c2=5/10=1/2. All ratios equal → infinitely many solutions (dependent). COMMON MISTAKES: Cross-multiplication: use the formula carefully with correct signs. Check consistency BEFORE solving — if inconsistent, no solution exists. Graphical method: use at least 3 points per line to verify.