Page 1: Introduction to Linear Equations

A linear equation in two variables x and y is of the form ax + by + c = 0, where a, b, c are real numbers and a, b not both zero.

Examples: 2x + 3y = 12, x - y + 5 = 0, 4x = 8 (here b=0).

Every solution (x, y) satisfying the equation represents a point on the line.

This chapter teaches how to solve, graph, and apply such equations.

Page 2: Solution of a Linear Equation

Any pair (x, y) that satisfies ax + by + c = 0 is a solution.

There are infinitely many solutions (corresponding to points on the line).

Example: For 2x + y = 5
If x=0 → y=5 (point (0,5))
If x=1 → y=3 (point (1,3))
If x=2 → y=1 (point (2,1))

Page 3: Graph of a Linear Equation

The graph of ax + by + c = 0 is always a straight line.

To plot: Find at least two points → join them.

Intercepts:

• x-intercept: Put y=0 → solve for x
• y-intercept: Put x=0 → solve for y

Plotting these two intercepts is the fastest method.

Page 4: Examples of Graphing

Page 5: Equations of Lines Parallel to Axes

x = a → vertical line (parallel to y-axis)

y = b → horizontal line (parallel to x-axis)

No intercepts needed — just draw accordingly.

Page 6: System of Two Linear Equations

Two equations together: a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0

Three possible cases:

• Intersecting lines → unique solution (consistent)
• Parallel lines → no solution (inconsistent)
• Coincident lines → infinitely many solutions (dependent)

Page 7: Graphical Method of Solution

Plot both lines.

• Intersection point → solution
• No intersection → no solution
• Lines overlap → infinite solutions

Page 8: Algebraic Methods - Substitution

Steps:

1. Express one variable in terms of other from one equation.
2. Substitute into second equation.
3. Solve for one variable.
4. Find second variable.

Page 9: Algebraic Methods - Elimination

Make coefficients of one variable equal → add/subtract.

Page 10: Cross-Multiplication Method

For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
x = (b₁c₂ - b₂c₁)/(a₁b₂ - a₂b₁)
y = (c₁a₂ - c₂a₁)/(a₁b₂ - a₂b₁)

Denominator zero → parallel/infinite cases.

Page 11: Consistency Conditions

• a₁/a₂ ≠ b₁/b₂ → unique solution
• a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → no solution
• a₁/a₂ = b₁/b₂ = c₁/c₂ → infinite solutions

Memorise these ratios!

Page 12: Key Formulas Summary

• General form: ax + by + c = 0
• x-intercept = -c/a, y-intercept = -c/b
• Parallel axes: x=a, y=b
• Consistency ratios
• Cross-multiplication formula

Page 13: Practice Questions - Easy (1-10)

1. Write 3 solutions of 2x + y = 7.
2. Find intercepts of 3x + 4y = 12.
3. Draw x = 4.
4. Solve x + y = 5, x = 3.
5. Is (2, -1) solution of 3x - 2y = 8?
6. Express y = 5 - 2x in ax + by + c = 0.
7. Find one solution of x - y = 0.
8. Graph y = 0.
9. Check if lines 2x + y = 4 and 4x + 2y = 8 are same.
10. Find k if x + ky = 5 has infinite solutions with x + 2y = 5.

Page 14: Practice Questions - Medium (11-20)

11. Solve graphically: 2x + y = 6, 4x + 2y = 12.
12. Solve by substitution: 3x - y = 3, 9x - 3y = 9.
13. Find k for no solution: 2x + ky = 8, 3x + 2y = 12.
14. Solve: x/2 + y/3 = 4, x/4 + y/6 = 3.
15. Graph x + 2y = 6 and find area with axes.
16. Solve elimination: 5x - 3y = 1, 2x + y = 7.
17. Check consistency: 4x + 6y = 12, 2x + 3y = 6.
18. Find equation of line through (0,3) and (4,0).
19. Solve cross-multiplication: 3x + 4y = 10, 2x - y = 5.
20. Word problem setup: Sum of ages.

Page 15: Practice Questions - Hard (21-30)

21. Solve: 2/x + 3/y = 13, 5/x - 4/y = -2 (substitution).
22. Find k for unique solution in general system.
23. Three equations word problem setup.
24. Graphically solve inconsistent system.
25. Find equation parallel to given line.
26. Reduce to linear: fractions.
27. Dependent system example.
28. Cross-multiplication with parameters.
29. Real-life application: distance-speed.
30. Prove infinite solutions graphically.

Page 16: Word Problems - Type 1 (Numbers)

Sum/difference of two numbers.

Let numbers be x, y → x + y = ..., x - y = ...

Solve → add/subtract.

Page 17: Word Problems - Type 2 (Ages)

Current ages, ratio, after/before years.

Always define variables clearly.

Page 18: Word Problems - Type 3 (Money/Items)

Cost of items, total amount.

Practice framing equations carefully.

Page 19: NCERT Exercise Types

Ex 3.1: Solutions and graphing

Ex 3.2: Algebraic solution methods

Ex 3.3: Consistency checking

Ex 3.4: Word problems

Page 20: Important Theorems Recap

• Graph is straight line
• Infinite solutions for one equation
• Three cases for pair of equations
• Ratio conditions for consistency

Page 21: Common Mistakes to Avoid

• Wrong intercepts calculation
• Sign errors in elimination
• Forgetting to check consistency
• Poor variable definition in word problems
• Plotting wrong points

Page 22: Previous Year Board Questions

Typical: Solve graphically (4 marks)

Algebraic solution (3 marks)

Word problem (4-5 marks)

Consistency check (2 marks)

Page 23: Exam Strategy Tips

• Draw neat graphs with scale
• Show all steps in algebraic methods
• Check solution by substitution
• Read word problems twice
• Memorise consistency ratios

Page 24: Quick Revision Formula Sheet

• ax + by + c = 0
• Unique: a1/a2 ≠ b1/b2
• No solution: ratios equal but ≠ c1/c2
• Infinite: all ratios equal
• Cross-multiplication method

Master these → full marks!

Page 25: Final Motivation

You've completed the 27-page Linear Equations guide!

This chapter is crucial for Coordinate Geometry ahead.

Practice graphing and word problems daily.

You're getting stronger every chapter 🦖

Keep going!