➕ Algebra 1 · Equations
Combining Like Terms
Simplify expressions like 3x + 5 + 2x − 1 by combining like terms: add the coefficients, keep the variable part, with diagrams, memory tricks and practice.
About standards and checking
Written by AI, then independently re-solved and checked by separate AI reviewers and automated answer checks. Written by Claude (Sonnet 5.5) from the concept template. Automated gate: schema, notation and every math answer verified by evaluated expressions. Three blind re-solves by separate AI solvers with no access to the answer key: every keyed answer matched. Two prose audits by a separate, stronger AI reviewer (Claude Opus 5.5): every finding fixed, no wrong answer keys found. Final small edits re-checked by the automated gate (no human teacher review yet). Last checked 2026-11-07.
- 6.EE.A.3: Apply the properties of operations to generate equivalent expressions.
- 7.EE.A.1: Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
- A-SSE.A.1a: Interpret parts of an expression, such as terms, factors, and coefficients.
By the end you can…
- Name the parts of an expression: terms, coefficients, constants and variable parts.
- Decide which terms are like terms and combine them by adding their coefficients.
- Handle the awkward cases: a lone letter (coefficient 1), negative and fractional coefficients, unlike powers, and a minus sign in front of parentheses.
Lesson (school style)
A term is one chunk of an expression, and like terms are chunks of the same kind. Combining like terms means adding up the chunks that match, so that an expression such as 3x + 5 + 2x − 1 becomes the shorter, equivalent expression 5x + 4. The big idea: add the coefficients and keep the variable part.
Terms, coefficients and constants
An expression is built from terms. A term is a number, a variable, or numbers and variables multiplied together. Terms are separated by plus and minus signs, and each term keeps the sign in front of it.
In 5p − 2q + 9, the terms are 5p, −2q and 9.
- The coefficient is the number multiplied by the variable part: 5 in 5p, and −2 in −2q.
- A constant is a term with no variable, like 9.
- A letter on its own has an invisible coefficient of 1. So p means 1p, and −p means −1p.
What makes terms “like”?
Like terms have exactly the same variable part: the same letters, each with the same exponent. The coefficients can be different.
- Like: 3s and −8s (both have the variable part s).
- Like: 2uv and 7vu (same letters; the order of multiplication does not matter).
- Like: 4 and −9 (constants are like terms with each other).
- Not like: 5s and 5s2 (s and s2 are different variable parts).
- Not like: 3s and 2u (different letters).
- Not like: 3s and 4 (one has a variable, the other does not).
Combining like terms never changes the value of an expression, whatever numbers the letters stand for.
Add the coefficients, keep the variable part
To combine like terms, add (or subtract) their coefficients and keep the variable part exactly as it is:
3s + 5s = (3 + 5)s = 8s
Three of something plus five of the same something is eight of it. This works for any coefficients, including negatives and fractions, because it is the distributive property read backward: 3s + 5s = (3 + 5)s.
The variable part is a label, not something you add. 3s + 5s = 8s, not 8s2. Adding like terms never changes the exponent. Multiplying can: 3s × 5s = 15s2.
Signs travel with their terms
Because each term keeps the sign in front of it, 9u − 4u means 9u + (−4u). The coefficients combine as 9 + (−4) = 5, so 9u − 4u = 5u. In 9 − 4u the second term is −4u, not 4u.
You may reorder terms to put like terms next to each other, as long as each sign moves with its term: 9 − 4u + 2 + u regroups as (9 + 2) + (−4u + u), which is 11 − 3u.
Unlike terms stay separate
Terms that are not alike cannot be merged, and that is fine. 3s + 2u is already as short as it gets, and so is s2 + s.
A quick test shows why s + s2 is not 2s2: at s = 3, we get s + s2 = 3 + 9 = 12, but 2s2 = 18.
Subtracting a group
A minus sign in front of parentheses applies to every term inside, so it flips each sign: −(3s − 5) = −3s + 5. Then combine as usual:
(7s + 2) − (3s − 5) = 7s + 2 − 3s + 5 = 4s + 7
The next lesson, on the distributive property, explains why this works.
Check with a test number
To check your work, replace the variable with a number in both the original and the simplified expression. The two values should match. Avoid 0, 1 and 2 as test numbers, because they can hide mistakes (for example, s and s2 are equal when s = 1, and s + s and s2 are equal when s = 2). One test cannot prove two expressions are always equal, but it catches most slips.
Key words
- Term
- One piece of an expression, separated from the others by plus or minus signs. It keeps the sign in front of it: in 5p − 2q + 9 the terms are 5p, −2q and 9.
- Coefficient
- The number multiplied by the variable part of a term. In −2q it is −2, and in p it is 1.
- Constant
- A term with no variable, like 9.
- Like terms
- Terms with exactly the same variable part: the same letters with the same exponents. Only like terms can be combined.
- Simplify
- To rewrite an expression in a shorter equivalent form, for example by combining like terms.
- Equivalent expressions
- Expressions that have the same value for every value of the variables, such as 3s + 5s and 8s.
Worked examples
Example 1. Simplify 3x + 5 + 2x − 1.
- List the terms with their signs: 3x, +5, +2x and −1.
- Group the like terms: the x-terms are 3x and 2x, and the constants are 5 and −1.
- Add the coefficients of the x-terms: 3x + 2x = (3 + 2)x = 5x.
- Combine the constants: 5 − 1 = 4.
- Write the result: 5x + 4.
- Check with x = 3: the original is 9 + 5 + 6 − 1 = 19 and the result is 15 + 4 = 19.
Answer: Answer: 5x + 4
Example 2. Simplify 4y − y + 7 − 9y.
- The lone y in −y has coefficient −1. The terms are 4y, −1y, +7 and −9y.
- Combine the y-terms: 4 − 1 − 9 = −6, so they make −6y.
- The only constant is 7.
- Write the result: −6y + 7.
- Check with y = 3: the original is 12 − 3 + 7 − 27 = −11 and the result is −18 + 7 = −11.
Answer: Answer: −6y + 7
Example 3. Simplify 5a + 3b − 2a + 4b − 6.
- There are three kinds of terms: a-terms, b-terms and a constant.
- The a-terms: 5a − 2a = 3a.
- The b-terms: 3b + 4b = 7b.
- The constant is −6. The a-terms, b-terms and constant are not like each other, so they stay separate.
- Write the result: 3a + 7b − 6.
- Check with a = 3 and b = 5: the original is 15 + 15 − 6 + 20 − 6 = 38 and the result is 9 + 35 − 6 = 38.
Answer: Answer: 3a + 7b − 6
Example 4. Simplify 2x2 + 5x − x2 + 3x − 4.
- The variable parts are x2 and x. These are different, so they are not like terms.
- The x2-terms: 2x2 − x2 = (2 − 1)x2 = x2.
- The x-terms: 5x + 3x = 8x.
- The constant is −4.
- Write the result, highest power first: x2 + 8x − 4.
- Check with x = 3: the original is 18 + 15 − 9 + 9 − 4 = 29 and the result is 9 + 24 − 4 = 29.
Answer: Answer: x2 + 8x − 4
Example 5. Simplify (2/3)t + 5 − (1/6)t − 1.
- The t-terms have fractional coefficients: (2/3)t and −(1/6)t.
- Use a common denominator: 2/3 = 4/6, so 4/6 − 1/6 = 3/6 = 1/2.
- So the t-terms combine to (1/2)t.
- The constants: 5 − 1 = 4.
- Write the result: (1/2)t + 4.
- Check with t = 6: the original is 4 + 5 − 1 − 1 = 7 and the result is 3 + 4 = 7.
Answer: Answer: (1/2)t + 4
Example 6. Simplify (6x + 5) − (2x − 3).
- The minus sign in front of the second group applies to both of its terms, so it flips both signs: −(2x − 3) = −2x + 3.
- Rewrite without parentheses: 6x + 5 − 2x + 3.
- Combine the x-terms: 6x − 2x = 4x.
- Combine the constants: 5 + 3 = 8.
- Check with x = 3: the original is (18 + 5) − (6 − 3) = 23 − 3 = 20 and the result is 12 + 8 = 20.
Answer: Answer: 4x + 8
Example 7. Simplify 3xy + 2x − xy + 5yx.
- The terms 3xy, −xy and 5yx all have the same letters, x and y, each to the first power. The order of multiplication does not matter, so 5yx is the same kind of term as xy.
- Combine their coefficients: 3 − 1 + 5 = 7, so they make 7xy.
- The term 2x has only the letter x, so it is not like the xy-terms and stays as it is.
- Write the result: 7xy + 2x.
- Check with x = 3 and y = 5: the original is 45 + 6 − 15 + 75 = 111 and the result is 105 + 6 = 111.
Answer: Answer: 7xy + 2x
In plain English
Think of a backpack. If it holds 3 notebooks and you add 5 more notebooks, you have 8 notebooks. If it holds 3 notebooks and you add 2 pens, you cannot say you have 5 notebooks or 5 pens. You have 3 notebooks and 2 pens.
Algebra works the same way. 3x + 5x is like 3 notebooks plus 5 notebooks, so it is 8x. The letter stays the same, and you add the numbers in front of it.
But 3x + 5 stays as it is. One part has an x and the other does not, like notebooks and pens. Also, x and x2 are different kinds, so they stay apart too.
One more thing: in algebra the letter is a number we do not know yet, not a real object. The rule is the same, and it works for every number the letter could be.
See it
The same idea, the fun way
Pick the one that clicks. Each one explains the same concept from a different angle.
Hangouts by friend
Last week you hung out with Maya 3 times and with Noor 2 times. This week you hung out with Maya 4 more times and with Noor 1 more time. To total it up, you count Maya hangouts with Maya hangouts and Noor hangouts with Noor hangouts.
How it connects: Like terms are counted together: 3m + 2n + 4m + n = 7m + 3n. Hangouts with Maya do not merge with hangouts with Noor, just as m-terms do not merge with n-terms. (In algebra the letter is an unknown number, not a friend, but the grouping rule is the same.)
The Sorting Hat
In the Harry Potter stories, a talking hat sorts each new student into one of four school houses. Every student ends up in exactly one house.
How it connects: Sorting terms works the same way: each term goes into the group that matches its variable part (the x group, the x2 group, the constants group). Then, inside each group, add the coefficients, keeping their signs: in 3x + 2x − x the x group gives 3 + 2 − 1 = 4, so 4x. Terms in different groups never merge, so x and x2 stay apart.
Sesame Street’s odd-one-out game
Sesame Street has a long-running game where you look at a group of items and spot the one that does not belong with the others.
How it connects: Try it on 3x, 5x, −x and 2x2. The odd one out is 2x2, because its variable part is x2 while the others have x. The three that match are like terms: 3x + 5x − x = 7x.
Birds of a feather flock together
The proverb means that similar people tend to gather with each other.
How it connects: Like terms flock together. In 7z + 3 + 2z, the two z-terms are the birds of a feather: 7z + 2z = 9z. The constant 3 stays on its own, so the result is 9z + 3.
Apples and oranges
Someone who is “comparing apples and oranges” is comparing two things that are too different to compare sensibly.
How it connects: In the same spirit, unlike terms do not merge. If a is the price of one apple and r is the price of one orange, then 3 apples and 2 oranges cost 3a + 2r, and that is as simple as it gets. You cannot write it as 5a or 5r. (The letters stand for numbers, here prices, not for the fruit itself.)
A taco-truck receipt
A receipt shows 5 tacos and 2 sodas. Later you add 3 more tacos and return 1 soda. Tacos go with tacos: 5 + 3 = 8. Sodas go with sodas: 2 − 1 = 1.
How it connects: Let t be the price of one taco and s the price of one soda. The bill is 5t + 2s + 3t − s = 8t + s. The returned soda is −s, which means −1s, so the sodas combine as 2s − 1s = 1s, written simply as s. A letter on its own has coefficient 1. (The letters are prices, which are numbers, not the food itself.)
A season record
A team’s record lists wins and losses separately. Weekend one: 3 wins and 1 loss. Weekend two: 2 wins and 2 losses.
How it connects: Let W stand for a win and L for a loss: (3W + 1L) + (2W + 2L) = 5W + 3L. Combine wins with wins and losses with losses. The record of 5 wins and 3 losses stays as two separate counts: it does not become 8W or 8L. (Here W and L are just labels for counting. In algebra a letter stands for a number, but the rule of combining only matching kinds is the same.)
Inventory stacks
In many video games, identical items stack in one inventory slot: 12 stone blocks and 20 more stone blocks make one stack of 32. A different item, such as dirt, goes in its own slot.
How it connects: Like terms stack: 12s + 20s = 32s. The coefficient is like the stack size, and the variable part is like the item type. Even s and s2 are different variable parts, so they take different slots. (In algebra the letter stands for a number, not an item, but only matching variable parts stack.)
Memory help
SAME
Same letters And Matching Exponents. Two terms are like terms only if they pass SAME. Constants have no letters, so any two constants pass.
The like-terms rhyme
Same letters, same exponents: they go together. Add the numbers in front, keep the letters whatever the weather. Different letters or exponents: leave them alone. Side by side they stay, each on its own.
Shapes and colors
Mark each kind of term with its own shape: circle the x-terms, box the constants, underline the x2-terms. Mark the sign inside the shape too. Then add up what is inside each shape.
The name-tag party
Picture a party where every guest wears a name tag showing a variable part: x, x2, y, or nothing for constants. Each guest also holds a number card, which is the coefficient with its sign (a guest with no number shown holds a 1). Guests with the same tag sit at the same table. At each table, add up the number cards (a negative card takes away), and that total is the new coefficient. Tables never merge, so x and x2 stay at separate tables.
The invisible 1
A lone x is really 1x, and −x is −1x. When you combine 4x − x, say it out loud: four x minus one x is three x.
Loop the sign with its term
Draw a loop around each term together with the sign on its left, such as the −4u in 9 − 4u. When you slide terms next to each other, the loops move with their signs.
Study notes
Key ideas
- A term is one chunk of an expression, separated by plus and minus signs. Each term keeps the sign in front of it.
- Like terms have exactly the same variable part: the same letters with the same exponents. The coefficients may differ. Constants are like terms with each other.
- To combine like terms, add the coefficients and keep the variable part: ax + bx = (a + b)x.
- Unlike terms stay separate. The simplified expression has the same value as the original for every value of the variable.
Steps
- Mark each term together with the sign in front of it.
- Group the like terms (same variable part), including the constants.
- Add the coefficients in each group. A lone letter has coefficient 1, and −x has coefficient −1.
- Write the result, usually with the highest power first and the constant last.
- Check with a test number such as 3 or 5 in the original and the result.
Formulas
- Combine rule: ax + bx = (a + b)x
- Minus before a group: −(a + b) = −a − b and −(a − b) = −a + b
- Cancelling: ax − ax = 0
Watch out
- A lone letter has coefficient 1: 9b − b = 8b, not 9.
- Only the coefficients are added. 3x + 5x = 8x, never 8x2. Adding never changes the exponent.
- x and x2 are not like terms. Neither are xy and x.
- The sign belongs to the term on its right: in 9 − 4y the term is −4y.
- A minus sign in front of parentheses flips every sign inside: −(2k − 3) = −2k + 3.
Why it works
- Why does it work? 3x + 5x = (3 + 5)x is the distributive property read backward: 3 of something plus 5 of the same something is 8 of it.
- Why can’t unlike terms combine? They are different quantities. At x = 3, x + x2 = 12 but 2x2 = 18, so the two expressions are not equal.
- Why may we rearrange terms? The order of addition does not matter, so you can slide terms next to their like terms as long as each sign moves with its term.
Common mistakes
Teach it back
Explain to a friend how to simplify 5n + 8 − 2n − 3 + n, and how you would check your answer with a test number.
Explaining it out loud shows what you really know. Did you…
- I sorted the terms into n-terms and constants.
- I remembered that the lone n has coefficient 1: 5n − 2n + n = 4n.
- I combined the constants, 8 − 3 = 5, and wrote 4n + 5.
- I checked with n = 3: both the original and the result equal 17.
Your 3-week memory plan
- TodayRead the lesson, then try the flashcards.
- TomorrowRecall it from memory first, then check.
- In 3 daysTake the true/false and practice questions.
- In 1 weekMix it with older topics and teach it back.
- In 3 weeksA last quick review before it fades.
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