What is a Linear Equation in Two Variables?
A linear equation in two variables is an equation that can be written in the form ax + by + c = 0, where a, b, and c are real numbers, and a and b are NOT both zero. The variables x and y have exponent 1 (they are linear).
Standard Form: ax + by + c = 0
Examples:
- 2x + 3y - 6 = 0 (a=2, b=3, c=-6)
- x - y = 5 (can be rewritten as x - y - 5 = 0)
- y = 2x + 1 (can be rewritten as 2x - y + 1 = 0)
Key Properties:
- A linear equation in two variables has infinitely many solutions
- Each solution is an ordered pair (x, y) that satisfies the equation
- When plotted on a coordinate plane, all solutions lie on a straight line
Why "Linear"?
The word "linear" comes from the Latin word "linea" meaning line. The graph of such an equation is always a straight line.
Real-Life Applications:
- Relationship between cost and quantity (C = mx + b)
- Distance-time relationship (d = rt)
- Budget constraints (2x + 3y = 100)