Lesson 7: Adding and subtracting terms

Introduction

The first six lessons were all about multiplying and dividing terms with indices making up a big part of this. In this (quite long) lesson, you're going to learn everything you need to know about adding and subtracting terms, which is completely different!

SECTION A

Adding and subtracting terms is also called collecting like terms because you can only add and subtract terms that are like.

Terms are like if the only thing that makes them different is the sign or the number in front of them (a number on its own can only be like another number on its own).

When you start, it may help to imagine terms as objects being put in (+) or taken out of (−) a shopping basket!

Examples

1. Simplify \(~a+a+a\)

Answer: \(~3a~\) (apple + apple + apple = 3 apples)

[Remember: \(~a\times a\times a=a^3~\). Told you! Completely different!]

2. Simplify \(~3b+b\)

Answer: \(~4b~\) (3 bananas + banana = 4 bananas)

3. Simplify \(~4c-2c+c\)

Answer: \(~3c~\) (4 carrots − 2 carrots + carrot = 3 carrots)

4. Simplify \(~2a+3b+5a+b\)

Answer: \(~7a+4b~\) (because \(~2a+5a=7a~\) and \(~3b+b=4b~\))

Note that only like terms have been combined. An apple and a banana does not make a banapple! It's just an apple and a banana. You can't simplify it any further.

5. Simplify \(~2a-3b+5a+b\)

Answer: \(~7a-2b~\) (because \(~2a+5a=7a~\) and \(~-3b+b=-2b~\))

This is REALLY important! There is no \(~3b~\) in this expression. Seriously, it's MINUS \(~3b~\). The sign to the left of EVERY term belongs to that term. This example is also a reminder of how important it is to be able to work with negative numbers.

If you're happy with these five examples, have a go at the practice questions before moving on to Section B.

Practise to master

SECTION A

Simplify each expression using your knowledge of collecting terms. For the first 16 questions, you won't have to worry about terms being LIKE.

01) \(~n+n~\)

\(2n\)

02) \(~d+d+d+d+d~\)

\(5d\)

03) \(~f+f+f+f~\)

\(4f\)

04) \(~5m+m~\)

\(6m\)

05) \(~6k+2k~\)

\(8k\)

06) \(~h+4h+7h~\)

\(12h\)

07) \(~2b+3b+b~\)

\(6b\)

08) \(~3t+t+4t+8t~\)

\(16t\)

09) \(~p+p-p+p+p-p~\)

\(2p\)
We now have minus signs aswell so we're also taking things out of our basket

10) \(~7x-x~\)

\(6x\)

11) \(~9y-2y~\)

\(7y\)

12) \(~6a+2a-3a~\)

\(5a\)

13) \(~-8c+12c+5c~\)

\(9c\)
No reason why the first term can't be negative but don't let that put you off. It is difficult to imagine 8 carrots being taken out of an empty basket! I did say this analogy would help you when you start, though.

14) \(~6e-11e+3e~\)

\(-2e\)
The answer can also be negative but that shouldn't bother you either

15) \(~-u+8u+3u-4u~\)

\(6u\)

16) \(~-15v+3v-9v+v~\)

\(-20v\)

Now you have to remember to collect LIKE terms only.

17) \(~4x+7y+2x~\)

\(6x+7y\)

18) \(~5s+2t-2s+3t~\)

\(3s+5t\)

19) \(~3x+4y-z-x+3y~\)

\(2x+7y-z\)

20) \(~14a+2b+a+7~\)

\(15a+2b+7\)
The number on its own is not LIKE any of the other terms

21) \(~2r+13+9r-6~\)

\(11r+7\)
The two numbers are LIKE each other and can be collected

22) \(~7e+2f-3+e-4f+5~\)

\(8e-2f+2\)

23) \(~7m+5n+2m-3n-m+p~\)

\(8m+2n+p\)

24) \(~5+9p+2q+3p-5q+1~\)

\(12p-3q+6\)

25) \(~8s-2t-1+3s-7t+5~\)

\(11s-9t+4\)

26) \(~6x+2y-4u-3v+3~\)

\(6x+2y-4u-3v+3\)
Can't be simplified as none of the terms are LIKE - sorry! Thought I'd throw that one in there.

27) \(~4c-2d-5c+3d-2~\)

\(-c+d-2\)
Or you might prefer \(~d-c-2~\) as it is nice to not have a leading minus sign ...

28) \(~2g-2h-5g-h+g~\)

\(-2g-3h\)
... though sometimes it's unavoidable!

SECTION B

We can make this a bit trickier by including terms with indices. By trickier I mean you have to concentrate a bit harder when deciding which terms are like.

Terms are like if the only thing that makes them different is the sign or the number in front of them. You already know this. In fact, I copied and pasted that sentence from Section A!

So, \(~5x~\) is NOT like \(~5x^2~\) is NOT like \(~2x^3~\), etc.
The first is an \(~x~\) term, the second is an \(~x\)-squared term and the third is an \(~x\)-cubed term.

Example 1

Simplify  \(2x+3x^2+5x+4x^2\)

Answer:  \(7x^2+7x\)

You can see that the two \(~x~\) terms have been collected and that the two \(~x^2~\) terms have been collected separately.

You'll also notice that the \(~x^2~\) term is written before the \(~x~\) term in the answer. Terms with higher powers tend to be written first. This is a good habit to get into.

Example 2

Simplify  \(5p+3q+6p^3-q^2+2p+q-5p^3-q^2\)

Answer:  \(p^3-2q^2+7p+4q\)

You could really do with having a question like this written on a piece of paper in front of you. Then you could cross off like terms as you collect them. Otherwise you might miss one out or count one twice. You should certainly do this in an exam.

OK, let's see if you've got it.

Practise to master

SECTION B

Simplify these expressions using your knowledge of collecting like terms with indices.

01) \(~x^2+2x+3x+1~\)

\(x^2+5x+1\)
Simplifying an expression with an \(~x^2~\) term, two \(~x~\) terms and a constant (number) term is something you'll find yourself doing every time you expand double brackets (Lesson 12).

02) \(~2x^2-7x+2x-9~\)

\(2x^2-5x-9\)

03) \(~-4x-x+3+5x^2~\)

\(5x^2-5x+3\)

04) \(~2x-3x^2+3x-6~\)

\(-3x^2+5x-6\)

05) \(~m^2+3n^2+2m^2-5n^2~\)

\(3m^2-2n^2\)

06) \(~8a^3-2b^3+5a^3+5b^3~\)

\(13a^3+3b^3\)

07) \(~6n^2+8n^3-3n^2~\)

\(8n^3+3n^2\)

08) \(~3u^2-2u^5+u^2+8u^5~\)

\(6u^5+4u^2\)

SECTION C

We're now going to include terms which consist of more than one letter. Once again, it becomes more difficult to decide which terms are like. Or does it? I'm gonna risk being annoying by copying and pasting THAT sentence again.

Terms are like if the only thing that makes them different is the sign or the number in front of them.

Example 1

Simplify  \(2gh+3g-5gh+g\)

Answer:  \(4g-3gh\)

Obviously a \(~g~\) term is NOT like a \(~gh~\) term!

Example 2

Simplify  \(3xy+2yx-xy\)

Answer:  \(4xy\)

Ah, this is important! Terms may still be like if they consist of the same letters in a different order ...

Example 3

Simplify  \(3xy^2+5yx^2-y^2x\)

Answer:  \(2xy^2+5x^2y\)

... but the powers must be on the same letters.

Confused? Read this ...

If the terms in the question were written with their letters in alphabetical order as they should be, this confusion would not arise! If you do get an expression like this, rewrite it with the letters in alphabetical order BUT make sure that any indices stay with their original letter.

So, in the third example, \(~3xy^2+5yx^2-y^2x~\) becomes \(~3xy^2+5x^2y-xy^2~\) and it's now much easier to see that the first and third terms are like.

In the second example, \(~3xy+2yx-xy~\) becomes \(~3xy+2xy-xy~\) and it's now clear that all three terms are like.

We're nearly there!

Practise to master

SECTION C

Simplify these expressions using your knowledge of collecting like multiple-letter terms.

01) \(~4ab+7ab-3ab~\)

\(8ab\)

02) \(~2xy+5xy-xy+2xy~\)

\(8xy\)

03) \(~6ab+3xy-4ab+xy~\)

\(2ab+4xy\)

04) \(~2pq-4p+6pq+8p~\)

\(8pq+4p\)

05) \(~3pq-5p+4pq-2q~\)

\(7pq-5p-2q\)

06) \(~7uv+2u-3uv+9~\)

\(4uv+2u+9\)

07) \(~5ab+2ba-ab~\)

\(6ab\)

08) \(~7xy+3yx-2xy+4yx~\)

\(12xy\)

09) \(~3abc-4bca+9cab~\)

\(8abc\)

10) \(~5xy^2+2y^2x-3yx^2~\)

\(7xy^2-3x^2y\)

11) \(~3a^2b+2ab^2-ba^2~\)

\(2a^2b+2ab^2\)

12) \(~7p^2q+pq^2-2q^2p~\)

\(7p^2q-pq^2\)

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