Algebra Basics II

By now you should understand what I mean when I say; algebraic letters, in mathematics, are ‘containers’. But now I am going to show you how we can do mathematics with these ‘containers’ (side note: each time I say the word ‘container’, or ‘letter’, I want you to think of the term ‘variable’).

If you’re happy understanding why something like; 5x + 7y  +  2x + 3y = 7x + 10y , works, then skip this post.

Lets say we have two people, Carl & Maxine, who go shopping. And let’s say they purchase the following items:

Carl:   5 apples, 3 carrots, 2 lemons

Maxine:   1 carrot, 12 apples, 9 lemons

A more simple way to express this information would be to reduce these words to algebraic letters, and express what each person bought as a sum:

Carl’s Purchase =   5a + 3c + 2l

Maxine’s Purchase =   1c + 12a + 9l

Convention: When a term is expressed as '5a' this is the same as
saying "5 lots of a" or "5 × a"
Convention: Usually we should express something like '1c' as just
'c'. As '1 × c' is still 'c'.

Now if we wanted to know what both of these shoppers bought, collectively, then all we need to do is add these two expressions together:

Items Bought = Carl’s Purchase + Maxine’s Purchase

    = 5a + 3c + 2l    +    c + 12a9l

He is where I would like to stop and introduce a new phrase; you can only do numerical operations on like terms!

Definition:    A Like Term is a term (number with one or more letters behind it e.g. 5a) which is similar (alike) to another term (has the same letter or letters behind it).

E.g.       3x   and   7x   are like terms because they both have the same letter/container/variable x at the end.

E.g.       I can add 3 apples and 2 apples to make 5 apples. But I cannot add 2 apples and 3 bananas, as they are different objects.

Hence, I can add 3a to 2a to make 5a. But the terms in the expression, 2a + 3b cannot be added, and cannot be simplified any further. So 2a and 3b are unlike terms.

So let’s continue our shopping scenario.

Items Bought = Carl’s Purchase + Maxine’s Purchase

    = 5a + 3c + 2l    +    c + 12a + 9l

    = 5a + 12a + 3c + c + 2l + 9l                                                                                                        (Here I have moved around the terms so that the like terms match. This proccess is usually called ‘collecting like terms‘, and most of the time we do it in our head)

    = 17a + 4c + 11l

So in total the Carl and Maxine collectively bought; 17 apples, 4 carrots, 11 lemons.

This basic pattern of collecting like terms and then performing addition, is the very first stepping stone for numeracy in algebra. The exact same method works for subtraction too. Practice a couple of questions below.

5x + 2x + 3y + y =
21a + 4b + 3a + b =
5a - 3a =
10f - 3f + 4g - g =
a + 2b - a - b =
10b -5a -5b + 15a =

If you found any of this confusing, please let me know and I’ll try my best to update it. In the mean time try watching a video or two on ‘like terms’. I’d start with this one.

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