Lesson 4 • 30 min read
Linear Equations
In This Lesson
1What is an Equation?
An equation is a mathematical statement that shows two expressions are equal, connected by an equals sign (=).
x + 5 = 12
Left side = Right side
The Balance Concept
Think of an equation like a balance scale. Whatever you do to one side, you must do to the other to keep it balanced.
✓ Keeps Balance
x + 5 = 12
x + 5 - 5 = 12 - 5
Subtract 5 from BOTH sides
✗ Breaks Balance
x + 5 = 12
x + 5 - 5 = 12
Only one side changed!
2Inverse Operations
Operations That Undo Each Other
To solve an equation, use the inverse (opposite) operation to isolate the variable.
Addition undoes Subtraction
Subtraction undoes Addition
Multiplication undoes Division
Division undoes Multiplication
Key Idea:
If a number is added to x, subtract it. If x is multiplied, divide. Do the opposite!
3Solving One-Step Equations
Addition/Subtraction Equations
Example 1: x + 5 = 12
x + 5 = 12
Subtract 5 from both sides
x + 5 - 5 = 12 - 5
x = 7
Example 2: y - 8 = 3
y - 8 = 3
Add 8 to both sides
y - 8 + 8 = 3 + 8
y = 11
Multiplication/Division Equations
Example 3: 3x = 15
3x = 15
Divide both sides by 3
3x ÷ 3 = 15 ÷ 3
x = 5
Example 4: n/4 = 6
n/4 = 6
Multiply both sides by 4
(n/4) × 4 = 6 × 4
n = 24
Equations with Negative Numbers
Example 5: x + (-3) = 7
x - 3 = 7
Add 3 to both sides
x - 3 + 3 = 7 + 3
x = 10
Example 6: -5x = 20
-5x = 20
Divide both sides by -5
-5x ÷ (-5) = 20 ÷ (-5)
x = -4
4Solving Two-Step Equations
The Two Steps
Undo Addition or Subtraction FIRST
Deal with the constant term
Undo Multiplication or Division SECOND
Deal with the coefficient
Remember: Work in REVERSE order of operations!
Example 1: Solve 2x + 3 = 11
2x + 3 = 11
Step 1: Subtract 3 from both sides
2x + 3 - 3 = 11 - 3
2x = 8
Step 2: Divide both sides by 2
2x ÷ 2 = 8 ÷ 2
x = 4
Example 2: Solve 5y - 7 = 18
5y - 7 = 18
Step 1: Add 7 to both sides
5y - 7 + 7 = 18 + 7
5y = 25
Step 2: Divide both sides by 5
5y ÷ 5 = 25 ÷ 5
y = 5
Example 3: Solve n/3 + 4 = 10
n/3 + 4 = 10
Step 1: Subtract 4 from both sides
n/3 + 4 - 4 = 10 - 4
n/3 = 6
Step 2: Multiply both sides by 3
(n/3) × 3 = 6 × 3
n = 18
5Checking Your Answer
Always Check Your Work!
Substitute your answer back into the original equation to verify it's correct.
Check: If 2x + 3 = 11, and x = 4
2(4) + 3 = 11
8 + 3 = 11
11 = 11 ✓ Correct!
If it doesn't check:
- Go back and redo your work
- Check for sign errors (+ vs -)
- Make sure you did the same operation on both sides