What Is a Linear Equation?

A linear equation is an equation where the highest power of the variable is 1. When graphed, it produces a straight line — hence the name. Linear equations are among the most important concepts in algebra and appear throughout science, finance, and engineering.

A typical linear equation looks like this: 2x + 5 = 13

The Golden Rule of Equations

Whatever you do to one side of an equation, you must do to the other side. This keeps the equation balanced, like a scale. Your goal is to isolate the variable (usually x) on one side.

Solving a One-Step Equation

These require just one operation to solve.

Example: x + 7 = 12

  1. Subtract 7 from both sides: x + 7 − 7 = 12 − 7
  2. Result: x = 5

Solving a Two-Step Equation

Example: 2x + 5 = 13

  1. Subtract 5 from both sides: 2x = 8
  2. Divide both sides by 2: x = 4

Always eliminate addition or subtraction before dealing with multiplication or division.

Solving Equations with Variables on Both Sides

Example: 5x − 3 = 2x + 9

  1. Move variable terms to one side: 5x − 2x = 9 + 3
  2. Simplify: 3x = 12
  3. Divide: x = 4

Solving Equations with Parentheses

Use the distributive property to expand brackets first.

Example: 3(x + 4) = 21

  1. Distribute: 3x + 12 = 21
  2. Subtract 12: 3x = 9
  3. Divide by 3: x = 3

Checking Your Answer

Always substitute your answer back into the original equation to verify it's correct. If both sides are equal, you've solved it!

Check x = 4 in 2x + 5 = 13: 2(4) + 5 = 8 + 5 = 13 ✓

Common Equation-Solving Mistakes

  • Forgetting to apply an operation to both sides of the equation.
  • Making sign errors when moving terms across the equals sign.
  • Distributing incorrectly (e.g., only multiplying the first term in parentheses).
  • Skipping the verification step.

Summary: Steps to Solve Any Linear Equation

  1. Simplify both sides (distribute, combine like terms).
  2. Move variable terms to one side.
  3. Move constant terms to the other side.
  4. Divide by the coefficient of the variable.
  5. Check your answer.

Linear equations are the gateway to all higher algebra. Once you're comfortable solving them confidently, systems of equations, inequalities, and functions all become much more manageable.