Combining Like Terms — Algebraic Expressions

💡 Explanation — What are like terms?
🍎 Using apples as an example:
3 apples + 5 apples = 8 apples
📐 In algebra:
\(3a + 5a\)
The two terms share the same variable (a) — they are like terms.
🔢 How do we simplify?
Add the coefficients: \(3 + 5 = 8\)
The variable stays: \(a\)
Answer: \(8a\)
✨ Golden rule: Like terms share exactly the same variable!
You can combine them by adding or subtracting their coefficients.

💡 Explanation — Why cannot this be simplified?
🍌🍊 Using fruit as an example:
4 bananas + 3 oranges = 4 bananas and 3 oranges.
You cannot say "7 somethings" — they are not the same thing!
📐 In algebra:
\(4b + 3o\)
b and o are different variables — these are unlike terms.
🚫 Important rule:
You can only combine terms with exactly the same variable!
✅ Like terms: \(3x + 5x\) ← same variable
❌ Unlike terms: \(3x + 5y\) ← different variables
💭 Think: you cannot add apples and bananas into one number!

💡 Explanation — Subtracting like terms
🥕 Using carrots as an example:
7 carrots − 3 carrots = 4 carrots
📐 In algebra:
\(7c - 3c\)
Both terms share the same variable (c) — they are like terms.
🔢 How do we subtract like terms?
Subtract the coefficients: \(7 - 3 = 4\)
The variable stays: \(c\)
Answer: \(4c\)
✨ Rule for subtraction:
Just like addition — subtract the coefficients!
\(7c - 3c = (7-3)c = 4c\)
💭 Remember: always check that the variable is identical before simplifying!

💡 Explanation — Different variables
🎈 Using balloons as an example:
5 red balloons + 2 blue balloons = 5 red and 2 blue (not 7 "somethings"!)
📐 In algebra:
\(5r + 2b\)
r (red) ≠ b (blue) — different variables — cannot combine!
🎨 Why not?
Red ≠ Blue → r ≠ b
Therefore you cannot add them into one number.
✅ Can simplify: \(5r + 3r\) ← same color!
❌ Cannot simplify: \(5r + 2b\) ← different colors!

💡 Detailed explanation
📦 Using boxes as an example:
8 large boxes − 5 large boxes = 3 large boxes
📐 In algebra:
\(8x - 5x\)
Both terms share the same variable (x = large size) — like terms.
🔢 Simplification process:
Step 1: Check — same variable? ✓ (both are x)
Step 2: Subtract coefficients: \(8 - 5 = 3\)
Step 3: Variable stays: \(x\)
✅ Answer: \(3x\)
💡 Tip: Always think about what the variable represents — if the items are the same type, they can be combined!

💡 Explanation
🚗 Using cars as an example:
6 red cars + 4 red cars = 10 red cars
🔴 Key concept:
All cars are the same type — red! So m represents "one red car".
📐 Simplification:
\(6m + 4m\)
Same variable (m) → like terms! ✓
Add coefficients: \(6 + 4 = 10\)
Answer: \(10m\)
💭 Compare:
If there were 6 red and 4 blue cars: \(6m + 4b\) ← cannot simplify!

💡 Advanced explanation
⚽🏀 Using balls as an example:
5 soccer balls + 3 basketballs = 5 soccer balls and 3 basketballs
(Not just "8 balls"!)
🎯 Why not?
• Soccer ball ≠ basketball
• Different size
• Different weight
• Different use
→ These are completely different things!
📐 In algebra:
\(5s + 3b\)
s ≠ b → different variables → cannot simplify!
✅ Can simplify: \(5s + 3s\)
❌ Cannot simplify: \(5s + 3b\) stays as is!

💡 Step-by-step explanation
📚 The scenario:
12 science books − 7 science books = 5 science books
🔍 Step 1: Identify like terms
\(12n - 7n\)
Both terms use n (one science book) → like terms! ✓
🔢 Step 2: Subtract coefficients
\(12 - 7 = 5\)
✏️ Step 3: Write the answer
Variable stays: n
Answer: \(5n\)
📖 Meaning:
n = one science book
5n = 5 science books
💭 Remember: you can always "check in plain language" — do the items match?

💡 Explanation
🌺🌸 Using flowers as an example:
9 roses + 6 sunflowers = 9 roses and 6 sunflowers
(Not "15 flowers" — they are not the same type!)
🌹 Why not the same type?
• A rose looks different from a sunflower
• They grow in different places
• Different smell
• Different shape
→ Completely different flowers!
📐 In algebra:
\(9r + 6s\)
r (rose) ≠ s (sunflower) → different variables → cannot simplify!
✅ Like terms: \(9r + 3r\) ← only roses
❌ Unlike terms: \(9r + 6s\) ← different types

💡 Detailed explanation with illustration
🍪 The scenario:
We started with 15 cookies in the jar.
We ate 9 cookies.
6 cookies remained.
📐 In algebra:
c = one cookie
\(15c\) = 15 cookies
\(9c\) = 9 cookies
🔢 Simplification:
\(15c - 9c\)
Same variable (c) → like terms! ✓
Subtract: \(15 - 9 = 6\)
Answer: \(6c\)
🎯 Meaning: 6c = 6 cookies remaining in the jar.
💭 Remember: all cookies are the same type = same variable = can simplify!

💡 Explanation
✏️ Using pencils as an example:
4 red pencils + 5 red pencils = 9 red pencils
🔴 Why can we simplify?
All pencils are the same — red! Same type = same variable.
📐 Simplification:
\(4p + 5p\)
Same variable (p) → like terms! ✓
Add coefficients: \(4 + 5 = 9\)
Answer: \(9p\)
p = one red pencil, so 9p = 9 red pencils.

💡 Musical explanation
🎵 Using music notes as an example:
10 high notes − 4 high notes = 6 high notes
🎼 The concept:
All notes are the same type — high notes! Same type = same variable.
📐 Simplification:
\(10n - 4n\)
Same variable (n) → like terms! ✓
Subtract: \(10 - 4 = 6\)
Answer: \(6n\)
💡 Note: If there were also low notes (different variable l), we could NOT combine them with n!

💡 Explanation
🧃🥤 Using drinks as an example:
8 juice bottles + 5 chocolate milk cartons = 8 juice and 5 chocolate milk
(Not "13 drinks" — different types!)
📐 In algebra:
\(8j + 5c\)
j (juice) ≠ c (chocolate milk) → different variables → cannot simplify!
✅ Can simplify: \(8j + 3j\) ← all juice
❌ Cannot simplify: \(8j + 5c\) ← different drinks

💡 Explanation
🎨 Using markers as an example:
20 markers − 12 markers = 8 markers
(All the same type — markers!)
🖍️ Why can we simplify?
All markers are one kind — same type = same variable.
📐 Simplification:
\(20m - 12m\)
Same variable (m) → like terms! ✓
Subtract: \(20 - 12 = 8\)
Answer: \(8m\)
Note: Even though some are used and some are not, they are all the same variable m!

💡 Explanation
⭐ Using stickers as an example:
11 star stickers + 7 star stickers = 18 star stickers
✨ Why can we simplify?
All stickers are the same — star stickers! Same type = same variable.
📐 Simplification:
\(11s + 7s\)
Same variable (s) → like terms! ✓
Add: \(11 + 7 = 18\)
Answer: \(18s\)

💡 Explanation
🎈🚩 Using decorations as an example:
6 balloons + 9 small flags = 6 balloons and 9 small flags
(Not "15 decorations" — different types!)
📐 In algebra:
\(6b + 9f\)
b (balloon) ≠ f (flag) → different variables → cannot simplify!
✅ Can simplify: \(6b + 4b\) ← all balloons
❌ Cannot simplify: \(6b + 9f\) ← different decoration types

💡 Explanation
🍕 The scenario:
We ate 3 slices + 5 slices remained = 8 slices at the start.
🔴 Why can we simplify?
All slices are from the same pizza — same type, same variable!
📐 In algebra:
\(3p + 5p\)
Same variable (p) → like terms! ✓
Add: \(3 + 5 = 8\)
Answer: \(8p\)
8p = 8 pizza slices total at the start.

💡 Explanation
🐟 Using fish as an example:
18 fish + 5 fish = 23 fish
(All fish in the aquarium — same type!)
🐠 Why can we simplify?
All are fish — same variable.
📐 Simplification:
\(18f + 5f\)
Same variable (f) → like terms! ✓
Add: \(18 + 5 = 23\)
Answer: \(23f\)
Note: Even though the fish were there at different times, they are all the same variable!

💡 Explanation
🎵 Using songs as an example:
14 songs − 6 songs = 8 songs
🎸 Why can we simplify?
All songs are the same type — same variable.
📐 Simplification:
\(14s - 6s\)
Same variable (s) → like terms! ✓
Subtract: \(14 - 6 = 8\)
Answer: \(8s\)

💡 Explanation — Summary question
📄 Using pages as an example:
25 blank pages + 17 filled pages = 25 blank and 17 filled
(Not "42 pages" — different types!)
In algebra:
e = one blank (empty) page
f = one filled page
\(25e + 17f\)
e ≠ f → different variables → cannot simplify!
✅ Can simplify: \(25e + 5e\) ← all blank
❌ Cannot simplify: \(25e + 17f\) ← different types

💡 Explanation — Partial simplification
🍎🍊 Identifying like terms:
• 5a (apples) ✓
• 3o (oranges) ✗
• 4a (more apples) ✓
🔍 Group by variable:
Apples: \(5a + 4a = 9a\)
Oranges: \(3o\) stays (different variable)
Simplification:
\(5a + 3o + 4a = 9a + 3o\)
✨ We combined the apples but kept the oranges separate — they are different variables!

💡 Explanation
🚗🏍️ Analysing the vehicles:
• 8c = 8 cars
• 5m = 5 motorcycles
• 3c = 3 more cars
🔍 Identify like terms:
Cars (c): \(8c + 3c = 11c\)
Motorcycles (m): stays as 5m
Simplification:
\(8c + 5m + 3c = 11c + 5m\)
✨ Cars combine; motorcycles stay separate — different variables!

💡 Explanation
📚 Analysing the books by topic:
• 12s = science books
• 7h = history books
• 5s = more science books
🔍 Identify like terms:
Science (s): \(12s + 5s = 17s\)
History (h): stays as 7h
Simplification:
\(12s + 7h + 5s = 17s + 7h\)
Science books combine; history books stay separate!

💡 Explanation — 4 terms
🎨 Analysing the art pieces:
• 6p = landscape paintings
• 9s = sculptures
• 4p = more landscapes
• 2s = more sculptures
🔍 Identify like terms:
Paintings (p): \(6p + 4p = 10p\)
Sculptures (s): \(9s + 2s = 11s\)
Simplification:
\(6p + 9s + 4p + 2s = 10p + 11s\)
✨ Two pairs of like terms — group and combine each pair separately!

💡 Explanation — Addition and subtraction
🌹🌻 Analysing the changes:
• 15r = roses at the start
• −7r = roses removed
• +8s = sunflowers added
🔍 Identify like terms:
Roses (r): \(15r - 7r = 8r\)
Sunflowers (s): stays as 8s
Simplification:
\(15r - 7r + 8s = 8r + 8s\)
Roses combine (addition and subtraction); sunflowers stay separate!

💡 Explanation — Complex expression
🎮 Analysing the games:
• 20a = action games (start)
• −8a = action games (deleted)
• +12p = strategy games (bought)
• +5a = action games (more bought)
🔍 Identify like terms:
Action (a): \(20a - 8a + 5a = 17a\)
Strategy (p): stays as 12p
Simplification:
\(20a - 8a + 12p + 5a = 17a + 12p\)
Combine all action-game terms; keep strategy separate!

💡 Explanation — Two pairs
🍪🍰 Analysing the baked goods:
• 18c + 9c = cookies
• 10k + 6k = cakes
🔍 Identify like terms:
Cookies (c): \(18c + 9c = 27c\)
Cakes (k): \(10k + 6k = 16k\)
Simplification:
\(18c + 10k + 9c + 6k = 27c + 16k\)
✨ Two pairs of like terms — simplify each pair separately!

💡 Explanation — Addition and subtraction
🎸🥁 Analysing the instruments:
• 4s = string instruments (added)
• 6p = percussion instruments (start)
• 3s = more string instruments
• −2p = percussion instruments (removed)
🔍 Identify like terms:
Strings (s): \(4s + 3s = 7s\)
Percussion (p): \(6p - 2p = 4p\)
Simplification:
\(4s + 6p + 3s - 2p = 7s + 4p\)
Two pairs — combine strings together, combine percussion together!

💡 Explanation — Expression with 5 terms
🐕🐈 Full analysis:
• 14d = dogs (start)
• 9c = cats (start)
• +6d = more dogs
• −5c = cats sold
• −3d = dogs sold
🔍 Identify like terms:
Dogs (d): \(14d + 6d - 3d = 17d\)
Cats (c): \(9c - 5c = 4c\)
Simplification:
\(14d + 9c + 6d - 5c - 3d = 17d + 4c\)
Group dogs together, group cats together — then simplify each!

💡 Explanation — Comprehensive summary question
🎓 Analysing the movement in class:
• 25s = students at the start
• −8s = students who went to recess
• +3t = teachers who came in (t ≠ s!)
• +5s = students who returned
🔍 Identify like terms:
Students (s): \(25s - 8s + 5s = 22s\)
Teachers (t): stays as 3t (different variable!)
Simplification:
\(25s - 8s + 3t + 5s = 22s + 3t\)
Students and teachers are different types — keep them separate!