Linear equations are the foundation of algebra and appear throughout mathematics, science, engineering, and everyday problem-solving. Learning to solve linear equations systematically gives you the skills to tackle more complex mathematical problems and real-world applications.
What Is a Linear Equation?
A linear equation contains variables raised to the first power only. The standard form is ax + b = c, where a, b, and c are numbers and x is the variable you're solving for.
Examples of linear equations:
2x + 5 = 13
3x - 7 = 8
x + 4 = 10
5x = 20
Basic Solving Strategy
The goal is to isolate the variable (x) on one side of the equation. Use inverse operations: if a number is added, subtract it; if multiplied, divide it.
The Golden Rule: Whatever you do to one side of the equation, do the same to the other side to keep it balanced.
Step-by-Step Examples
Example 1: Simple Linear Equation
Problem: 2x + 5 = 13
Step 1: Subtract 5 from both sides
2x + 5 - 5 = 13 - 5
2x = 8
Step 2: Divide both sides by 2
2x รท 2 = 8 รท 2
x = 4
Check: 2(4) + 5 = 8 + 5 = 13 โ
Example 2: Equation with Subtraction
Problem: 3x - 7 = 8
Step 1: Add 7 to both sides
3x - 7 + 7 = 8 + 7
3x = 15
Step 2: Divide both sides by 3
3x รท 3 = 15 รท 3
x = 5
Check: 3(5) - 7 = 15 - 7 = 8 โ
Example 3: Variables on Both Sides
Problem: 5x + 3 = 2x + 12
Step 1: Subtract 2x from both sides
5x - 2x + 3 = 2x - 2x + 12
3x + 3 = 12
Step 2: Subtract 3 from both sides
3x + 3 - 3 = 12 - 3
3x = 9
Step 3: Divide both sides by 3
x = 3
Check: 5(3) + 3 = 15 + 3 = 18; 2(3) + 12 = 6 + 12 = 18 โ
Common Linear Equation Types
| Form | Example | Solution |
|---|---|---|
| ax = b | 4x = 20 | x = 5 |
| ax + b = c | 3x + 5 = 14 | x = 3 |
| ax - b = c | 2x - 8 = 6 | x = 7 |
| ax + b = cx + d | 5x + 2 = 2x + 8 | x = 2 |
| a(x + b) = c | 3(x + 2) = 15 | x = 3 |
Equations with Fractions
Example:
Problem: (x + 3)/2 = 5
Step 1: Multiply both sides by 2
2 ร (x + 3)/2 = 2 ร 5
x + 3 = 10
Step 2: Subtract 3 from both sides
x + 3 - 3 = 10 - 3
x = 7
Equations with Decimals
Example:
Problem: 0.5x + 1.2 = 3.7
Step 1: Subtract 1.2 from both sides
0.5x = 3.7 - 1.2
0.5x = 2.5
Step 2: Divide by 0.5 (or multiply by 2)
x = 2.5 รท 0.5
x = 5
Negative Numbers and Signs
Example:
Problem: -3x + 4 = 16
Step 1: Subtract 4 from both sides
-3x = 16 - 4
-3x = 12
Step 2: Divide by -3 (remember: dividing by negative flips nothing for x)
x = 12 รท (-3)
x = -4
Check: -3(-4) + 4 = 12 + 4 = 16 โ
Distributive Property
When multiplying across parentheses, distribute to every term:
a(b + c) = ab + ac
Example: 2(x + 3) = 10
2x + 6 = 10
2x = 4
x = 2
Real-World Applications
Linear equations solve practical problems:
Example: Salary Calculation
You earn $15 per hour plus a $50 weekly bonus.
If you earn $200 in a week, how many hours did you work?
15h + 50 = 200
15h = 150
h = 10 hours
Example: Distance Problem
You drive 60 mph. After 2 hours, you're 30 miles behind schedule.
What distance were you supposed to travel?
60(2) = 120 miles traveled
120 + 30 = 150 miles planned
Tips for Success
- Simplify both sides first (combine like terms)
- Get variables on one side, numbers on the other
- Use inverse operations in reverse order of operations
- Always check your answer by substituting back
- Be careful with negative signs and distributive property
No Solution vs All Numbers
Some equations have no solution (the variable cancels out to false), while others are true for all values of x.
No solution: 2x + 3 = 2x + 5 (simplifies to 3 = 5, false)
All solutions: 2(x + 1) = 2x + 2 (simplifies to identity)
Use our Linear Equation Solver to instantly solve equations and verify your work.