Lesson · Foundations of Algebra · Building expressions

Combining Like Terms

Two terms can be added into one exactly when their variable parts are identical, and the licence to do it is the distributive law read backwards. The rule is short; the work is in seeing which terms qualify and in keeping the signs attached while you collect.

Open the interactive lesson Practice

3x + 4y − x + 2y = 2x + 6y collected 3xy + 2yx − xy = 4xy the order of the letters is irrelevant 5x²y − 2xy² already finished x²y and xy² are not alike 3x² + 5x² = 8x² NOT 8x⁴ the exponent is untouched

Three that combine, one that does not

Core ideas

What this lesson establishes

Collecting like terms is the only simplification a sum offers, so the question of which terms are alike is the whole lesson.

Like terms have identical variable parts

The same letters, each to the same exponent. 3x² and −7x² are alike; 3x² and 3x³ are not, and no rearrangement will make them so. Both conditions, every time.

The coefficients have no say in it

100x³ and x³ are like terms. x³ and x² are not, however close they look. Whether two terms may be combined is decided entirely by the letters, and the numbers are what you then do arithmetic on.

Combining like terms is the distributive law backwards

3x² + 5x² = (3 + 5)x² = 8x² is ac + bc = (a + b)c with c = x². That is why no new rule is needed, and it is also why unlike terms cannot be combined: there is no common c to take out.

The material

Which terms may be added

The definition, the property that licenses it, and the four ways two terms fail to be alike.

Definition · Like terms

Two terms are like terms when their variable parts are identical: the same variables, each raised to the same exponent.

5x²y and −2x²y are like terms. 5x²y and 5xy² are not: the letters match but the exponents are attached to different ones. Constant terms are all alike, since each has an empty variable part.

The order in which the letters are written does not affect anything, because multiplication is commutative (“Properties of the Real Numbers”). 3xy and 2yx have the same variable part written two ways, and they are like terms. Sorting the letters alphabetically inside every term is a cheap habit that makes this visible instead of a judgement call.

Theorem · Why like terms combine

For any numbers a and b and any variable part v, av + bv = (a + b)v.

This is the distributive law ac + bc = (a + b)c of “The Distributive Law”, read from right to left, with c taken to be the variable part v.

The law holds for all real numbers, and v stands for a real number whatever value the letters in it take, so the statement holds for every value of every variable involved.

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3x + 4y − x + 2y x terms : 3x + (−x) = (3 − 1)x = 2x y terms : 4y + 2y = (4 + 2)y = 6y answer : 2x + 6y

The −x contributes −1, not −0 and not +1. This is the invisible coefficient from “Variables, Expressions, and Terms” doing real work: 3 − 1 = 2, and a reader who treats −x as contributing nothing gets 3x instead.

Four ways two terms fail to be alike

Different exponent on the same letter. x² and x³. There is nothing to add: x² + x³ is already as short as it gets.
Different letters. 4x and 4y. Equal coefficients are irrelevant, and 4x + 4y is finished.
The same letters with the exponents swapped. 5x²y and 2xy². At x = 2, y = 3 the first is 60 and the second is 36, so they are not even close to interchangeable.
One inside a bracket that has not been expanded. 3x + 2(x + 1) has no two like terms as it stands. Distribute first, getting 3x + 2x + 2, and then there are.

That last case is the ordering rule for the whole subject: expand, then collect. Collecting first is not wrong so much as premature, because the terms you need are still hidden inside the bracket.

Like terms are not a polynomial-only idea. 3√2 + 5√2 = 8√2 for exactly the same reason, with √2 playing the part of the variable part — it is one fixed number, whatever it is. And √2 + √3 cannot be combined, for exactly the same reason x + y cannot.

Example · Two collections, in full

4a − 3b + 2a − b + 7 collects to 6a − 4b + 7: the a terms give 4 + 2 = 6, the b terms give −3 − 1 = −4, and the 7 has nothing to join. 2x + 3 − 5 − 2x collects to −2: the x terms give 2 − 2 = 0 and vanish, and 3 − 5 = −2 remains.

Interactive

Like terms

Same letters, same exponents, or they are not like

Method

Collecting like terms

Step 1 is skipped more often than it is done, and step 1 is what makes the rest safe.

Expand every bracket first

Terms inside a bracket are not available to be collected. Distribute, using “The Distributive Law”, and only then look at what you have.

Attach every sign to the term that follows it

Rewrite the expression as a sum of signed terms if that helps. A term carried across the page without its sign is the commonest way a correct grouping produces a wrong answer.

Group by variable part, sorting the letters within each term

Write the letters of each term in alphabetical order, then gather the terms whose variable parts are now literally identical. Terms with no letters form their own group.

Add the coefficients within each group, and leave the variable part alone

The exponent never changes: 3x² + 5x² is 8x². If a group's coefficients sum to zero, that whole group disappears from the answer.

Alike or not

Score 0 / 0

Choose the answer you can justify.

One expression, grouped

4a − 3b + 2a − b + 7, with each group written out.

signs attached: 4a + (−3b) + 2a + (−b) + 7 a terms : 4a + 2a = (4 + 2)a = 6a b terms : −3b + (−b) = (−3 − 1)b = −4b constants: 7 7 answer 6a − 4b + 7 check at a = 1, b = 1: original 4 − 3 + 2 − 1 + 7 = 9 answer 6 − 4 + 7 = 9

The b group is where the arithmetic is. −b contributes −1, so the sum is −3 − 1 = −4 and not −3 − 0 = −3 or −3 + 1 = −2. Both of those wrong answers are produced by the same cause: the invisible 1 was not read.

The check tests a = 1, b = 1 only, which is enough to catch a dropped term or a wrong sign and is not a proof of equality. What proves it is the theorem above, which holds for every value of a and b.

For a faded collection, group 6a²b − a²b + 3ab² without touching the coefficients until the groups are marked. Only the first two terms join, so the result is 5a²b + 3ab². Combining all three means the swapped exponents were treated as though their variable parts matched.

Common mistakes

Adding the exponents instead of the coefficients3x² + 5x² = 8x². Exponents add when powers are multiplied, which is “Integer Exponents”. Adding like terms leaves the variable part exactly as it was, and only the coefficients change.

Writing 3x + 4y = 7xyThere is nothing to combine: the variable parts differ, so no common factor can be taken out. Testing at x = 1, y = 1 gives 7 on both sides and proves nothing; testing at x = 2, y = 1 gives 10 against 14.

Dropping the sign of a subtracted termIn 5x² + 2x − 3x² + x the third term is −3x², so the x² group is 5 − 3 = 2, not 5 + 3 = 8. Attach every sign before grouping and this cannot happen.

Check your understanding

✓

Finish when you can say instantly whether two terms may be added, and why.

Expand and collect 3(2x − y) − x + 4y − 7, marking every group by its variable part before adding coefficients. Justify each merge by the distributive law read backwards and explain why an unlike term stays. Linear Equations and Inequalities collects on each side of an equation and preserves every sign error made here.

Note: This is the last of the three reading skills. “Variables, Expressions, and Terms” found the terms, “The Distributive Law” got them out of their brackets, and this lesson decides which may be merged. Polynomials and Factoring does the same three things to polynomials and calls it addition.