Linear Equations in Two Variables

🎯 What are Linear Equations?

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The highest power of the variable is 1.

General Form: ax + by = c
Example: 2x + 3y = 6

📊 One Variable vs Two Variables

One Variable: Has only one unknown (x)

Example: 2x + 5 = 11
Solution: x = 3

Two Variables: Has two unknowns (x and y)

Example: 2x + 3y = 12
Solutions: Multiple pairs (x, y)

📈 Graphical Representation

Linear equations in two variables can be represented as straight lines on a coordinate plane.

Slope-Intercept Form: y = mx + b
Where: m = slope, b = y-intercept

Every point on the line represents a solution to the equation.

♾️ Types of Solutions

Unique Solution: Lines intersect at one point

No Solution: Parallel lines (never meet)

Infinite Solutions: Same line (coincident lines)

System:
2x + 3y = 6
4x + 6y = 12
Result: Infinite solutions

🔧 Methods to Solve

1. Substitution Method: Solve one equation for one variable, then substitute

2. Elimination Method: Add or subtract equations to eliminate a variable

3. Graphical Method: Plot both equations and find intersection point

Example: x + y = 5, x - y = 1
Solution: x = 3, y = 2

💡 Real-World Applications

Linear equations are used in:

Economics: Supply and demand curves
Physics: Motion with constant velocity
Engineering: Circuit analysis
Business: Cost and revenue analysis