Adding and taking away like terms

In the last les­son you learned to sort terms into like and unlike. Now you put that sort­ing to work. Adding and tak­ing away like terms is how long alge­bra expres­sions are made short and tidy. You will use it when you solve equa­tions, work out perime­ters with let­ters, and sim­plify for­mu­las in later classes.

Think of a closed box of sweets. You have not counted them yet.

A let­ter can stand for a num­ber. It can stand for dif­fer­ent num­bers at dif­fer­ent times.

Call the num­ber of sweets in one box $a$. One box may hold $10$ sweets. The next box may hold $20$.

You never need to know that num­ber here. You only need to count boxes.

Two boxes hold $2a$ sweets. Three boxes hold $3a$ sweets.

Now put all of those boxes on the table. You have five boxes. So you have $5a$ sweets.

You did not open a sin­gle box. You still knew the answer.

Terms and expres­sions

A piece of maths writ­ten with let­ters and num­bers is called an expres­sion.

Here is one: $2a + 3b$.

Each part that is added or taken away is a term. So $2a + 3b$ has two terms. They are $2a$ and $3b$.

The let­ter part of a term is the let­ter or let­ters in it. You leave the num­ber in front out. In $5x$ the let­ter part is $x$.

Some terms have two let­ters. One box holds $a$ sweets, so $b$ boxes hold $a \times b$ sweets. You write that as $ab$.

So $ab$ means $a$ times $b$. In $3ab$ the let­ter part is $ab$.

Remem­ber. An expres­sion is made of terms. A term has a num­ber in front and a let­ter part.

What a like term is

Terms with the same let­ter part are like terms. They are also called sim­i­lar terms.

So $2a$ and $3a$ are like terms. Both are built from $a$.

$a$ and $2a$ are like terms too. The let­ter part of each one is $a$.

But $2a$ and $3b$ are not like terms. One counts $a$, the other counts $b$.

A plain num­ber and a let­ter are unlike as well. So $2$ and $a$ are unlike terms.

A plain num­ber has no let­ter part at all. The term $a$ has one. So the two can­not be joined.

This mat­ters because you can only add things of one kind. You can­not add boxes of sweets to bags of rice. You can only say how many you have of each.

Remem­ber. Like terms have the same let­ter part. Only like terms can be joined.

Adding like terms

Two $a$s and three more $a$s make five $a$s. So $2a + 3a$ is $5a$.

2a + 3a

$$\begin{aligned}&2a + 3a \\ &= (2 + 3)a \\ &= 5a\end{aligned}$$

Look at what changed. Only the num­ber in front changed. The let­ter part stayed as $a$.

That is the whole idea. You are count­ing how many $a$s you have. You are not chang­ing what $a$ is.

The num­ber in front

In $5x$ there are two parts mul­ti­plied together. They are $5$ and $x$.

The $5$ is called the co-effi­cient of $x$. In $-6xy$ the $-6$ is the co-effi­cient of $xy$.

On this page you only need the num­ber in front. That is the num­ber you add or take away.

A let­ter on its own has a co-effi­cient of $1$. One box is still one box. So $a$ means $1a$, and $y$ means $1y$.

a + a

$$\begin{aligned}&a + a \\ &= 1a + 1a \\ &= 2a\end{aligned}$$

Here are more sums of like terms. Each one adds the num­bers and keeps the let­ter part.

In the last row the let­ter part is $ab$. It is longer, but the idea does not change. Three $ab$s and four $ab$s make seven $ab$s.

Remem­ber. You can­not add $2a$ and $3b$. The let­ter parts are dif­fer­ent, so leave them as they are.

Tak­ing away like terms

Tak­ing away works the same way.

Start with seven $x$s. Take three $x$s away. Four $x$s are left.

7x - 3x

$$\begin{aligned}&7x - 3x \\ &= (7 - 3)x \\ &= 4x\end{aligned}$$

Again the let­ter part did not move. You only worked out $7 - 3$.

The order mat­ters when you take away. In a writ­ten sub­trac­tion, start with the term on the left. Then remove the term on the right.

So $9a - 4a$ is $5a$. But $4a - 9a$ is not $5a$.

Start with four $a$s and take nine away. You go below zero. That answer is $-5a$.

When noth­ing is left

Some­times you take away every­thing you had.

5a - 5a

$$\begin{aligned}&5a - 5a \\ &= (5 - 5)a \\ &= 0a \\ &= 0\end{aligned}$$

Five $a$s taken from five $a$s leaves none.

Write $0$, and not $0a$. Zero boxes hold noth­ing, what­ever is inside a box. So zero of any­thing is just zero.

When the answer goes below zero

Some­times you take away more than you started with.

3a - 7a

$$\begin{aligned}&3a - 7a \\ &= (3 - 7)a \\ &= -4a\end{aligned}$$

You know that $3 - 7$ is $-4$. So the answer is $-4a$.

You are still count­ing $a$s. Only the count went below zero. So the minus sign sits with the count. The $a$ has not changed.

You had three $a$s. You had to give away seven. So you gave away the three you had, and you still owe four more.

All eight together

Here are eight of these, includ­ing the ones worked out above. Cover the answers and try them first.

An answer can have two terms

It is easy to think every answer must be one term. It does not have to be.

The expres­sion $2a + 3b$ can­not be made shorter.

You do not know what $a$ and $b$ stand for. They may well be dif­fer­ent num­bers. So you can­not count them together.

The short­est true answer keeps both terms. Mak­ing it shorter would make it wrong.

Leav­ing two terms is not a fail­ure. It is the right answer.

Remem­ber. If the let­ter parts are dif­fer­ent, the terms stay apart. An answer with two terms is still a fin­ished answer.

The order of the work

1. Look at the let­ter part of each term.
2. Check whether the two let­ter parts are the same.
3. If they are the same, add or take away the num­bers in front.
4. Keep the let­ter part exactly as it was.
5. If they are dif­fer­ent, leave both terms as they are.

Longer expres­sions

Real expres­sions often have more than two terms. Use the same order of work, one group at a time. Each term keeps the sign writ­ten in front of it.

Exam­ple 1. Sim­plify $4x + 3y + 2x$

Step 1: The let­ter parts are $x$, $y$ and $x$. Step 2: $4x$ and $2x$ are like terms. $3y$ has no part­ner. Step 3: Join the $x$ terms and leave $3y$ alone.

$$\begin{aligned}4x + 3y + 2x &= 4x + 2x + 3y \\ &= (4 + 2)x + 3y \\ &= 6x + 3y\end{aligned}$$

Exam­ple 2. Sim­plify $5a + 2b - 3a + 6b$

The $a$ terms are $5a$ and $-3a$. The $b$ terms are $2b$ and $6b$. Notice that the minus sign trav­els with $3a$.

$$\begin{aligned}5a + 2b - 3a + 6b &= 5a - 3a + 2b + 6b \\ &= (5 - 3)a + (2 + 6)b \\ &= 2a + 8b\end{aligned}$$

Exam­ple 3. Sim­plify $7mn - 2m + 3 - 9mn + 5m$

There are three groups: $mn$ terms, $m$ terms, and a plain num­ber.

$$\begin{aligned}7mn - 2m + 3 - 9mn + 5m &= 7mn - 9mn - 2m + 5m + 3 \\ &= (7 - 9)mn + (-2 + 5)m + 3 \\ &= -2mn + 3m + 3\end{aligned}$$

The answer has three terms. None of them can be joined, so it is fin­ished.

Prac­tice

Make each expres­sion as short as you can.

Two of them can­not be made shorter. Their answers are the same as the ques­tions.

Check your work below.

Remem­ber. Add or take away the num­bers in front. Keep the let­ter part unchanged. Leave unlike terms alone.

Com­mon mis­takes

• Adding the let­ters as well as the num­bers. $2a + 3a$ is $5a$, not $5a^{2}$ or $5aa$.
• Join­ing unlike terms. $7p + 4q$ is not $11pq$; it stays as it is.
• For­get­ting the hid­den $1$. $8k - k$ is $7k$, not $8$.
• Writ­ing $0a$ instead of $0$ when every­thing is taken away.
• Swap­ping the order in a take-away. $2y - 5y$ is $-3y$, not $3y$.
• Los­ing a minus sign when terms are moved around. The sign in front of a term moves with it.

Key terms

Expres­sion

A piece of maths writ­ten with let­ters and num­bers, such as $2a + 3b$.

Term

Each part of an expres­sion that is added or taken away.

Let­ter part

The let­ter or let­ters in a term, leav­ing out the num­ber in front.

Co-effi­cient

The num­ber in front of the let­ter part; in $-6xy$ it is $-6$.

Like terms

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

Sim­plify

Make an expres­sion as short as it can go by join­ing like terms.

Answers

1. $a + a = 2a$
2. $4a + 5a = 9a$
3. $2x + 9x = 11x$
4. $6y + y = 7y$
5. $3ab + 5ab = 8ab$
6. $7m + 8m = 15m$
7. $9a - 4a = 5a$
8. $6x - 6x = 0$
9. $2y - 5y = -3y$
10. $10p - 3p = 7p$
11. $4ab - 9ab = -5ab$
12. $8k - k = 7k$
13. $7p + 4q$ can­not be made shorter; the let­ter parts dif­fer.
14. $3a + 2$ can­not be made shorter; $2$ has no let­ter part.