🔣 Simplifying Algebraic Expressions
Cambridge Lower Secondary · Grade 7 · Algebra
Collecting Like Terms
3x + 2y + 5x − y = 8x + y
Group x terms, group y terms
Multiplying Terms
2x × 3y = 6xy
Multiply coefficients, join letters
Index Laws
x² × x³ = x⁵ · x⁶ ÷ x² = x⁴
Add powers when multiplying, subtract when dividing
Watch terms combine! 🧱
What you'll learn:
- Writing algebraic expressions from words
- Identifying like terms (same letter and same power)
- Collecting like terms: 3x + 5x = 8x
- Simplifying with multiple variables: 3x + 2y + 5x − y = 8x + y
- Multiplying terms: 3 × 2x = 6x, 2x × 3y = 6xy
- Dividing terms: 6x ÷ 2 = 3x
- Index laws: x² × x³ = x⁵, x⁶ ÷ x² = x⁴
📖 Learn: Simplifying Expressions
Part 1: Writing Algebraic Expressions
An algebraic expression uses letters (variables) to represent unknown numbers. We write operations using standard notation.
📌 "3 more than n" → n + 3
📌 "5 times x" → 5x (we don't write the × sign)
📌 "y divided by 4" → y/4 or y ÷ 4
📌 "double a then subtract 7" → 2a − 7
💡 A coefficient is the number in front of a letter: in 5x, the coefficient is 5. If there is no number, the coefficient is 1 (e.g. x means 1x).
Part 2: Identifying Like Terms
Like terms have exactly the same letter(s) and power(s). Only like terms can be added or subtracted.
| Term |
Like terms with it |
NOT like terms |
| 3x |
5x, −2x, x, 10x |
3y, 3x², 3, xy |
| 4x² |
−x², 7x², x² |
4x, 4y², 4 |
| 2xy |
5xy, −xy |
2x, 2y, 2x²y |
💡 Constants (plain numbers like 4, −7, 100) are all like terms with each other.
Part 3: Collecting Like Terms
To simplify, group like terms together and add/subtract their coefficients.
Step 1: Identify all like terms in the expression
Step 2: Add/subtract coefficients within each group
Step 3: Write the simplified expression
Example: 3x + 2y + 5x − y
x terms: 3x + 5x = 8x · y terms: 2y − y = y
Answer: 8x + y
Example: 4a + 3b − 2a + 5b − 1
a terms: 4a − 2a = 2a · b terms: 3b + 5b = 8b · constants: −1
Answer: 2a + 8b − 1
💡 The sign (+ or −) belongs to the term that follows it. Always carry the sign with the term!
Part 4: Multiplying and Dividing Terms
Multiplying: multiply the coefficients and write the letters side by side.
3 × 2x = 6x (3 × 2 = 6, keep x)
2x × 3y = 6xy (2 × 3 = 6, x × y = xy)
4x × 2x = 8x² (4 × 2 = 8, x × x = x²)
Dividing: divide the coefficients and cancel letters.
6x ÷ 2 = 3x (6 ÷ 2 = 3, keep x)
12xy ÷ 4x = 3y (12 ÷ 4 = 3, xy ÷ x = y)
💡 For dividing, think of it as a fraction: 12xy/4x → cancel 4 into 12 → 3, cancel x → 3y
Part 5: Index Laws
When multiplying or dividing powers of the same base letter, use these laws:
| Law |
Rule |
Example |
| Multiply |
xᵃ × xᵇ = xᵃ⁺ᵇ (add powers) |
x² × x³ = x⁵ |
| Divide |
xᵃ ÷ xᵇ = xᵃ⁻ᵇ (subtract powers) |
x⁶ ÷ x² = x⁴ |
| Power of 1 |
x¹ = x |
x³ × x = x³ × x¹ = x⁴ |
Example: 3x² × 4x³ = (3 × 4)(x² × x³) = 12x⁵
Example: 10x⁵ ÷ 2x² = (10 ÷ 2)(x⁵ ÷ x²) = 5x³
💡 Only apply index laws when the base letters are the same! x² × y³ ≠ (xy)⁵
💡 Worked Examples
Example 1: Writing Expressions from Words
"A bag has n sweets. Mia takes 4 out, then triples what remains. Write an expression."
Step 1: After taking 4 out: n − 4
Step 2: Triple it: 3 × (n − 4) = 3(n − 4)
Expanded: 3n − 12
📝 "Triples" means multiply by 3. "Halves" means divide by 2.
Example 2: Collecting Like Terms
Simplify: 5x + 3y − 2x + 4y − 6 + 1
Step 1: Identify groups — x terms: 5x, −2x | y terms: 3y, 4y | constants: −6, 1
Step 2: x terms: 5x − 2x = 3x
Step 3: y terms: 3y + 4y = 7y
Step 4: constants: −6 + 1 = −5
Answer: 3x + 7y − 5
Example 3: Multiplying Terms
Simplify: 3x × 4y and 5a × 2a
3x × 4y: Multiply coefficients: 3 × 4 = 12. Join letters: x × y = xy. Answer: 12xy
5a × 2a: Multiply coefficients: 5 × 2 = 10. Apply index law: a × a = a². Answer: 10a²
💡 Always multiply the numbers first, then deal with the letters separately.
Example 4: Index Laws in Action
Simplify: 2x³ × 5x² and 8x⁵ ÷ 4x²
2x³ × 5x²:
· Coefficients: 2 × 5 = 10
· Index law (multiply → add): x³ × x² = x³⁺² = x⁵
Answer: 10x⁵
8x⁵ ÷ 4x²:
· Coefficients: 8 ÷ 4 = 2
· Index law (divide → subtract): x⁵ ÷ x² = x⁵⁻² = x³
Answer: 2x³
🧱 Algebra Tile Visualizer
Click the buttons below to add algebra tiles to the grid. Watch the expression auto-simplify as you build it!
Click buttons above to add tiles · Click a tile to remove it
Expression: (empty)
🎮 Sort the Terms!
Terms will fall from above. Click a term, then click the correct bin to score a point! Be quick — there's a timer!
Selected term shows in yellow. Click a bin to sort it.
🎚️ Coefficient Slider
Drag the sliders to change a and b, and watch ax + bx simplify live!
3x + 2x
simplifies to
5x
✏️ Exercise 1: Identifying Like Terms
Select the option that lists ALL and ONLY the like terms for the highlighted term.
✏️ Exercise 2: Collecting Like Terms
Simplify each expression by collecting like terms.
✏️ Exercise 3: Multiplying & Dividing Terms
Simplify each expression. Remember: multiply/divide coefficients, then deal with letters.
✏️ Exercise 4: Index Laws
Apply index laws to simplify. Add powers when multiplying, subtract when dividing.
✏️ Exercise 5: Mixed Simplification
Mix of all skills. Simplify each expression fully.
🏆 Challenge Questions
These questions require deeper thinking. Some have multiple steps.