| --- |
| source: ncert |
| topic: algebra |
| class_level: class_10 |
| chapter: ch3 |
| difficulty: intermediate |
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| |
| 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. |
