Introduction to Linear Equations
Learn to solve one-step and two-step equations using inverse operations — the foundation of all algebra.
What is a Linear Equation?
A linear equation is a mathematical statement that shows two expressions are equal, using an equals sign (=). It contains a variable (like x) that we need to find. The goal is to find the value of the variable that makes the equation true.
The Balance Model
Think of an equation like a set of balance scales. Whatever you do to one side, you must do to the other side to keep it balanced.
x + 3
10
Subtract 3 from both sides → x = 7
Golden Rule of Equations:
Whatever operation you perform on one side, you must perform the same operation on the other side.
One-Step Equations
A one-step equation requires only one inverse operation to solve. Use the opposite operation to isolate the variable.
Addition → use Subtraction
x + 5 = 12
x = 12 − 5
x = 7
Subtraction → use Addition
x − 4 = 9
x = 9 + 4
x = 13
Multiplication → use Division
3x = 18
x = 18 ÷ 3
x = 6
Division → use Multiplication
x⁄4 = 5
x = 5 × 4
x = 20
Two-Step Equations
A two-step equation requires two inverse operations. Always undo addition/subtraction first, then undo multiplication/division.
Solve: 2x + 3 = 11
Subtract 3 from both sides: 2x + 3 − 3 = 11 − 3 → 2x = 8
Divide both sides by 2: 2x ÷ 2 = 8 ÷ 2 → x = 4
Check: 2(4) + 3 = 8 + 3 = 11 ✓
Key Vocabulary
Equation
A mathematical statement where two expressions are equal, connected by an equals sign (=).
Variable
A letter (often x or n) that represents an unknown value we are trying to find.
Inverse Operation
The opposite operation. Addition and subtraction are inverses. Multiplication and division are inverses.
Solution
The value of the variable that makes the equation true. Always check by substituting back into the original equation.
Worked Examples
Solve: x + 8 = 15
Step 1: Subtract 8 from both sides: x = 15 − 8
x = 7
Check: 7 + 8 = 15 ✓
Solve: 5n − 4 = 21
Step 1: Add 4 to both sides: 5n = 21 + 4 = 25
Step 2: Divide both sides by 5: n = 25 ÷ 5
n = 5
Check: 5(5) − 4 = 25 − 4 = 21 ✓
Solve: m⁄3 + 2 = 7
Step 1: Subtract 2 from both sides: m⁄3 = 5
Step 2: Multiply both sides by 3: m = 5 × 3
m = 15
Check: 15 ÷ 3 + 2 = 5 + 2 = 7 ✓
Knowledge Check
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Key Concepts Summary
- ●An equation has an equals sign and shows that two expressions have the same value.
- ●Use inverse operations to isolate the variable — whatever you do to one side, do to the other.
- ●For two-step equations, undo addition/subtraction first, then undo multiplication/division.
- ●Always check your solution by substituting the answer back into the original equation.
- ●Think of the balance model — keep both sides equal at every step.