A linear equation is an equation in which the highest power of the variable(s) is 1. This means the variable is not squared, cubed, or multiplied by another variable. Linear equations are called linear because their graphs form a straight line.

👉 A linear equation always includes an equals sign (=) and shows that two expressions are equal.

Forms of Linear Equations

1. Linear Equation in One Variable

This has one variable (like $x$).

General form:

ax + b = c

where a, b, c are numbers and a≠0a \neq 0a=0.

Example:

3x + 5 = 14

Solution:

3x = 9 ⇒ x = 3

The solution is the value of the variable that makes the equation true.

2. Linear Equation in Two Variables

This has two variables (usually $x$ and $y$).

General form:

ax + by = c

Example:

2x + 3y = 6

This equation has infinitely many solutions. Each solution is an ordered pair (x, y) that makes the equation true. When graphed, all solutions lie on a straight line.

Key Characteristics of Linear Equations

• Variables have an exponent of 1

• No variables are multiplied together

• The graph is a straight line

• Can have:

  • One solution (one variable)

  • Infinitely many solutions (two variables)

  • No solution (in special cases)

Examples vs. Non-Examples

✅ Linear Equations

• 4x + 7 = 15

• 2x – y = 5

❌ Not Linear

• x^2 + 3 = 7(variable is squared)

• xy = 6 (variables multiplied)

Why Are Linear Equations Important?

• They model real-life situations (distance, money, time, speed)

• They are the foundation of algebra

• They help predict and compare relationships between quantities