What Is a Linear Equation?
A linear equation is an algebraic equation where the highest power of the variable is 1. When graphed, it forms a straight line. They appear in virtually every area of mathematics and real life — from budgeting to physics. Mastering linear equations is one of the most valuable steps in your math journey.
The general form is: ax + b = c, where x is the unknown and a, b, c are numbers.
The Golden Rule of Equation Solving
Whatever you do to one side of the equation, you must do to the other side. This keeps the equation balanced. Think of it like a scale — add weight to one side, and you must add the same to the other.
Step 1: One-Step Equations
These require a single operation to isolate the variable.
- x + 5 = 12 → Subtract 5 from both sides: x = 7
- x − 3 = 10 → Add 3 to both sides: x = 13
- 4x = 20 → Divide both sides by 4: x = 5
- x/6 = 3 → Multiply both sides by 6: x = 18
Step 2: Two-Step Equations
These require two operations. Always undo addition/subtraction before multiplication/division.
Example: 3x + 4 = 19
- Subtract 4 from both sides: 3x = 15
- Divide both sides by 3: x = 5
Always check your answer by substituting back: 3(5) + 4 = 15 + 4 = 19 ✓
Step 3: Multi-Step Equations
These may involve expanding brackets and combining like terms before isolating the variable.
Example: 2(x + 3) − 4 = 12
- Expand the bracket: 2x + 6 − 4 = 12
- Simplify: 2x + 2 = 12
- Subtract 2: 2x = 10
- Divide by 2: x = 5
Step 4: Variables on Both Sides
Move all variable terms to one side and constants to the other.
Example: 5x − 3 = 2x + 9
- Subtract 2x from both sides: 3x − 3 = 9
- Add 3 to both sides: 3x = 12
- Divide by 3: x = 4
Common Mistakes to Avoid
| Mistake | Example | Fix |
|---|---|---|
| Not applying operation to both sides | x + 3 = 7 → x = 7 | x = 7 − 3 = 4 |
| Sign errors when moving terms | x − 4 = 6 → x = 6 − 4 | x = 6 + 4 = 10 |
| Incorrect bracket expansion | 2(x + 3) = 2x + 3 | 2(x + 3) = 2x + 6 |
| Skipping the check step | Assuming answer is correct | Always substitute back |
Practice Strategy
Work through each type in order — don't jump to variables-on-both-sides until one-step equations feel effortless. Aim for 10–15 practice problems per type before progressing. Speed and accuracy will come with repetition.