Like and unlike terms

Alge­bra lets you work with num­bers you do not know yet. Before you can add or take away such num­bers, you must know which ones belong together. This les­son teaches that sort­ing: which terms are like terms, which are unlike, and how to tell at a glance. You will use it in every alge­bra les­son that fol­lows.

Pic­ture a small box with sweets inside. The lid is shut. You can­not see in. You do not know how many sweets are there.

You can still talk about that num­ber. Give it a short name. Call it $x$. Now $x$ stands for the num­ber of sweets in the box.

Tomor­row the box may hold a dif­fer­ent num­ber. The name still works. $x$ means what­ever is in the box that day.

A let­ter used this way has a name. It is called a vari­able. A vari­able is a sym­bol that can stand for many dif­fer­ent num­bers.

A plain num­ber is not like that. $5$ is always $5$. A fixed value like that is called a con­stant.

The two parts of a term

A term is one small piece of maths. It can be a num­ber on its own, like $6$. It can be a let­ter on its own, like $x$. It can be num­bers and let­ters mul­ti­plied together, like $6x$. It can be them divided too, like $\displaystyle \frac{x}{2}$.

Most terms have two parts. One part is a num­ber. One part is a let­ter. In $2a$ the num­ber part is $2$ and the let­ter part is $a$.

The num­ber part has a name. It is called the co-effi­cient. In $2a$ the co-effi­cient is $2$.

The let­ter part has a name too. It is called the lit­eral part. Watch that part closely. It tells you what the term is count­ing. That is what decides whether two terms will join.

Like terms

Two terms are like terms when their lit­eral parts are the same. The num­bers in front can be dif­fer­ent. That does not mat­ter.

Think about fruit for a moment. $2$ apples and $3$ apples are the same kind of thing. You can push them into one pile. The pile holds $5$ apples.

Now look at $2a$ and $3a$. $2a$ means two lots of $a$. $3a$ means three lots of $a$. Both are count­ing $a$s. So they are like terms.

$a$ is still a num­ber. Two lots of $a$ and three lots of $a$ are lots of the same num­ber. So they pile up the way apples do. $b$ is a dif­fer­ent num­ber. Lots of $b$ will not join lots of $a$.

$a$ and $2a$ are like terms as well. $a$ means one lot of $a$. You could write $1a$, but nobody does. The $1$ is left out because one lot is the ordi­nary case. So $a$ is count­ing $a$s too.

Remem­ber. Like terms have another name. Some books call them sim­i­lar terms. The two names mean the same thing.

Unlike terms

Two terms are unlike terms when their lit­eral parts are not the same.

Go back to the fruit. $2$ apples and $3$ bananas are not the same kind of thing. You can put them in one bowl. You still can­not call the bowl five apples. You can­not call it five bananas either.

$2a$ and $3b$ are the same story. One term counts $a$s. The other counts $b$s. They are unlike terms.

$2$ and $a$ are unlike terms too. The $2$ is just two. Two of what? It does not say. But $a$ counts $a$s. Two on its own and two $a$s are not the same kind of thing. So they can­not join.

Terms with two let­ters

Some terms have two let­ters. In $xy$ the lit­eral part is $xy$. That is not the same as $x$ on its own. $x$ counts $x$s. $xy$ counts $xy$s. So $5x$ and $5xy$ are unlike terms.

The order of the let­ters does not mat­ter. $ab$ means $a \times b$, and $ba$ means $b \times a$. Those are the same num­ber. So $3ab$ and $7ba$ are like terms. Many teach­ers ask you to write the let­ters in alpha­bet­i­cal order, so that a match is easy to see.

Pow­ers count as part of the lit­eral part too. $x^{2}$ means $x \times x$. That is not the same as $x$. So $4x$ and $4x^{2}$ are unlike terms.

The four exam­ples side by side

Remem­ber. The lit­eral part decides, because it says what is being counted. The num­ber in front only says how many. So the num­ber in front never decides.

Check­ing a pair of terms

Are $4x$ and $9x$ like terms? Cover the num­bers with your fin­ger. Then look at what is left.

Two terms, one test

$$\begin{aligned}&\text{The literal part of } 4x \text{ is } x \\ &\text{The literal part of } 9x \text{ is } x \\ &\text{The literal parts match, so the terms are like terms}\end{aligned}$$

More worked exam­ples

Exam­ple 1. Are $7p$ and $p$ like terms?

Step 1: Cover the num­bers. $7p$ leaves $p$. The term $p$ has no num­ber shown, which means $1p$, so it also leaves $p$. Step 2: Both lit­eral parts are $p$. So they are like terms.

Exam­ple 2. Are $6mn$ and $6m$ like terms?

Step 1: The lit­eral part of $6mn$ is $mn$. The lit­eral part of $6m$ is $m$. Step 2: $mn$ and $m$ are dif­fer­ent. So they are unlike terms, even though both start with $6$.

Exam­ple 3. Sort the terms of $3x + 5y + 2x + 4 + y + 1$.

Step 1: List each term with its lit­eral part: $3x$ has $x$, $5y$ has $y$, $2x$ has $x$, $4$ has none, $y$ has $y$, $1$ has none.

Step 2: Group the matches. The $x$ group is $3x$ and $2x$. The $y$ group is $5y$ and $y$. The plain num­bers are $4$ and $1$. That gives three groups of like terms.

Why this mat­ters

Like terms can be joined into one term. $2a$ and $3a$ both count $a$s. Put them together and you get $5a$.

Unlike terms can­not be joined. $2a$ and $3b$ count dif­fer­ent things. The pile has no sin­gle name. So $2a + 3b$ has to stay as it is.

That is why you sort the terms first. You find the ones that match. You join those, and you leave the rest alone. The next les­son does that work.

Prac­tice

Say whether each pair is like or unlike. Look at the lit­eral parts only.

Com­mon mis­takes

• Look­ing at the num­bers in front. $4y$ and $4z$ share a $4$ but are unlike; $2b$ and $11b$ have dif­fer­ent num­bers but are like.
• Think­ing a let­ter on its own has no num­ber. $k$ means $1k$, so $10k$ and $k$ are like terms.
• Treat­ing $x$ and $xy$ as a match because both con­tain $x$. The whole lit­eral part must be the same.
• Treat­ing $x$ and $x^{2}$ as like terms. The pow­ers are part of the lit­eral part.
• Pair­ing a plain num­ber with a let­ter term, such as $8$ with $8t$.

Key terms

Vari­able

A let­ter or sym­bol that can stand for many dif­fer­ent num­bers.

Con­stant

A fixed value, such as $5$, that never changes.

Term

One piece of an expres­sion: a num­ber, a let­ter, or num­bers and let­ters mul­ti­plied or divided.

Co-effi­cient

The num­ber part of a term; in $2a$ it is $2$.

Lit­eral part

The let­ter part of a term; it says what is being counted.

Like terms

Terms with the same lit­eral part, also called sim­i­lar terms.

Unlike terms

Terms whose lit­eral parts are not the same.

Answers

1. $3x$ and $7x$: like, both lit­eral parts are $x$.
2. $4y$ and $4z$: unlike, one lit­eral part is $y$, the other is $z$.
3. $m$ and $6m$: like, both lit­eral parts are $m$.
4. $9$ and $9p$: unlike, $9$ has no lit­eral part.
5. $2b$ and $11b$: like, both lit­eral parts are $b$.
6. $8t$ and $8$: unlike, $8$ has no lit­eral part.
7. $10k$ and $k$: like, both lit­eral parts are $k$.
8. $5x$ and $5xy$: unlike, one lit­eral part is $x$, the other is $xy$.