Chapter Summary - Linear Equations

Linear Equations Mastery

Review all three methods and learn when to use each one!

Three Solution Methods

📊

Graphical

Plot lines to find intersection. Best for visualization.

🔄

Substitution

Express one variable, substitute. Best for simple equations.

➖

Elimination

Make coefficients equal, eliminate. Best for complex problems.

All three methods solve the same problems - choose based on convenience!

Ratio Test for Solution Types

For equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0

Unique Solution

a₁/a₂ ≠ b₁/b₂

Lines intersect
Consistent

No Solution

a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Lines parallel
Inconsistent

Infinite Solutions

a₁/a₂ = b₁/b₂ = c₁/c₂

Lines coincide
Dependent

Problem-Solving Strategy

1 Read carefully and identify unknowns

2 Form equations from conditions

3 Choose appropriate method

4 Solve systematically

5 Verify your solution

6 Interpret in problem context

Real-World Applications

👨‍👧

Age & Time Problems

💰

Financial Calculations

🔢

Number Problems

📐

Geometric Problems

🛒

Commerce & Trade

🚗

Distance & Speed

Linear equations model countless real-world scenarios!

Method Comparison

Method

Best For

Advantage

Limitation

Graphical

Visualization

See solution

Not precise

Substitution

Simple forms

Direct

Can be messy

Elimination

Equal coeffs

Clean steps

Needs planning

Special Cases Recognition

Unique Solution

Get specific values

x = a, y = b
One answer

Infinite Solutions

0 = 0 (true)

Dependent
Many answers

No Solution

0 = k (false)

Inconsistent
Impossible