Page 1: Introduction
Pair of linear equations: a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Solutions: unique, infinite, or none.
Graphical: intersecting, coincident, parallel lines.
Page 2: Graphical Method
- Intersecting lines → unique solution (consistent)
- Coincident lines → infinite solutions (dependent)
- Parallel lines → no solution (inconsistent)
Page 3: Algebraic Methods Overview
- Substitution
- Elimination
- Cross-multiplication
Choose based on coefficients.
Page 4: Substitution Method
2. Substitute in second equation.
3. Solve for one variable.
4. Find second variable.
Page 5: Elimination Method
Best when coefficients easy to make equal.
Page 6: Cross-Multiplication Method
Direct formula for solution.
Page 7: Consistency Conditions
a₁/a₂ = b₁/b₂ = c₁/c₂ → infinite
a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → no solution
Page 8: Word Problems
Form two equations from statement.
Common: ages, numbers, speed, cost, mixture.
Page 9: Key Points Summary
- Three cases of solution
- Ratio conditions
- All three algebraic methods
- Graphical interpretation
Page 10: Practice Questions - Easy (1-10)
- Consistent pair example.
- Ratio for no solution.
- Solve 2x + y = 5, x + y = 4.
- Graph intersecting lines.
- Substitution simple problem.
- Elimination easy coefficients.
- Infinite solutions condition.
- Cross-multiplication for standard form.
- Parallel lines ratio.
- Unique solution ratio.
Page 11: Practice Questions - Medium (11-20)
- Solve by substitution: 3x + 2y = 11, 2x + 3y = 4.
- Elimination: 5x - 3y = 11, -10x + 6y = -22.
- Find k for infinite solutions: 2x + 3y = 7, (k+1)x + 9y = 21.
- Word problem: two numbers sum 27, difference 3.
- Check consistency: 2x + 3y = 12, 4x + 6y = 24.
- Cross-multiplication numerical.
- Ages problem: father twice son, sum 60.
- Speed problem: upstream downstream.
- Cost problem: pens and pencils.
- Graphical solution description.
Page 12: Practice Questions - Hard (21-30)
- Find k for no solution.
- Complex word problem: mixture.
- Three variables reduction to two.
- Graphical + algebraic verification.
- Fractional coefficients problem.
- Speed distance time advanced.
- Investment interest problem.
- Board level word problem.
- Consistency proof using ratios.
- Multiple methods same problem.
Page 13: Consistency Ratio Table
All three cases with examples.
Page 14: Common Mistakes
- Wrong ratio comparison
- Sign errors in elimination
- Incorrect variable expression in substitution
- Missing cross-multiplication denominator
- Wrong equation formation in word problems
Page 15: Previous Year Questions
Word problems, consistency, algebraic solution.
Page 16: Exam Tips
- Check consistency first in some questions
- Show all steps in algebraic methods
- Draw neat graph if asked
- Read word problems twice
- Verify solution by substitution
Page 17: Quick Revision Sheet
All methods, conditions, formulas.
Page 18: Final Motivation
Chapter 3 complete! Pair of equations is scoring with practice.
Master word problems.
Class 10 Maths unstoppable 🦖
Page 19: Extra Numericals
More solved word problems.
Page 20: Method Comparison
When to use which method.
Page 21: Word Problem Types
Ages, numbers, speed, cost.
Page 22: Consistency Cases
Detailed examples.
Page 23: Graphical Interpretation
Lines and solutions.
Page 24: Cross-Multiplication Derivation
Step-by-step.
Page 25: Board Pattern Questions
Common types.
Page 26: Extra Practice
More questions.
Page 27: Thank You & Copyright
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