Solving Simple Equations
Master the balance principle and inverse operations for the ParaPro test
What You'll Learn
Understanding Equations
The foundation of algebra
Expressions vs. Equations
Expression
3x + 5
- • No equals sign
- • Can be simplified
- • Cannot be solved
Equation
3x + 5 = 14
- • Has an equals sign
- • Can be solved
- • Has a specific solution
The Balance Principle
An equation is like a balanced scale. Whatever you do to one side, you must do to the other side to keep it balanced.
x + 3 = 7
Subtract 3 from both sides:
x + 3 - 3 = 7 - 3
x = 4
Inverse Operations
Operations that undo each other
Addition ↔ Subtraction
x + 5 = 12
→ Subtract 5 from both sides
x = 7
x - 3 = 8
→ Add 3 to both sides
x = 11
Multiplication ↔ Division
3x = 15
→ Divide both sides by 3
x = 5
x/4 = 2
→ Multiply both sides by 4
x = 8
Solving One-Step Equations
Use one inverse operation
Addition Equations
Solve: x + 9 = 15
Step 1: To isolate x, subtract 9 from both sides
x + 9 - 9 = 15 - 9
x = 6
Check: 6 + 9 = 15 ✓
Subtraction Equations
Solve: x - 7 = 11
Step 1: To isolate x, add 7 to both sides
x - 7 + 7 = 11 + 7
x = 18
Check: 18 - 7 = 11 ✓
Multiplication Equations
Solve: 4x = 28
Step 1: To isolate x, divide both sides by 4
4x ÷ 4 = 28 ÷ 4
x = 7
Check: 4 × 7 = 28 ✓
Division Equations
Solve: x/5 = 3
Step 1: To isolate x, multiply both sides by 5
(x/5) × 5 = 3 × 5
x = 15
Check: 15 ÷ 5 = 3 ✓
Solving Two-Step Equations
Work backwards through operations
The Order Matters!
Two-Step Process:
- Step 1: Undo addition or subtraction first
- Step 2: Undo multiplication or division second
Think of it like unwrapping a present, remove the outer layer first!
Example 1: Basic Two-Step
Solve: 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
Check: 2(4) + 3 = 8 + 3 = 11 ✓
Example 2: With Subtraction
Solve: 3x - 5 = 16
Step 1: Add 5 to both sides
3x - 5 + 5 = 16 + 5
3x = 21
Step 2: Divide both sides by 3
3x ÷ 3 = 21 ÷ 3
x = 7
Check: 3(7) - 5 = 21 - 5 = 16 ✓
Example 3: With Division
Solve: x/2 + 4 = 9
Step 1: Subtract 4 from both sides
x/2 + 4 - 4 = 9 - 4
x/2 = 5
Step 2: Multiply both sides by 2
(x/2) × 2 = 5 × 2
x = 10
Check: 10/2 + 4 = 5 + 4 = 9 ✓
Checking Your Solutions
Always verify your answer
The Substitution Check
Always check your answer by substituting it back into the original equation:
Check: Is x = 5 correct for 3x - 2 = 13?
- 1. Substitute 5 for x: 3(5) - 2 = ?
- 2. Calculate: 15 - 2 = 13
- 3. Compare: 13 = 13 ✓
Since both sides are equal, x = 5 is correct!
Common Mistakes to Avoid
Not doing the same to both sides
x + 5 = 12 → x = 12 - 5
✓ x + 5 - 5 = 12 - 5 → x = 7
Wrong order in two-step equations
2x + 6 = 14 → x + 6 = 7 (dividing first)
✓ Subtract first, then divide: 2x = 8 → x = 4
Sign errors with negatives
-x = 5 → x = 5
✓ -x = 5 → x = -5 (or multiply both sides by -1)
Not checking the solution
✓ Always substitute your answer back to verify!
Practice Problems
Test your understanding
One-Step Equations
x + 13 = 25
y - 8 = 15
5m = 35
n/4 = 9
Two-Step Equations
2x + 7 = 19
3y - 4 = 11
x/5 + 2 = 7
4x - 1 = 15
Show Answers
One-Step Equations:
1. x = 12 (subtract 13)
2. y = 23 (add 8)
3. m = 7 (divide by 5)
4. n = 36 (multiply by 4)
Two-Step Equations:
5. x = 6 (2x = 12, then x = 6)
6. y = 5 (3y = 15, then y = 5)
7. x = 25 (x/5 = 5, then x = 25)
8. x = 4 (4x = 16, then x = 4)
Key Takeaways
Balance Principle: Whatever you do to one side, do to the other
Inverse Operations: Use opposite operations to isolate the variable
Two-Step Order: Undo addition/subtraction first, then multiplication/division
Always Check: Substitute your answer back to verify it's correct
Continue Learning
Ready to Practice?
Test your equation solving skills with practice questions
