Lesson 1: Introducing substitution & formulae

Algebra starts here!

I'm going to assume that you have little or no knowledge of algebra at this point but that you've been doing number work since you started school. I'm assuming that you can add, subtract, multiply and divide; that you have come across indices (powers and roots); that you have some experience with negative numbers; that you've worked with fractions and decimals; and that you've heard about the order of operations (sometimes called BIDMAS or BODMAS). You need to be aware that the better your number skills, the further you'll be able to progress with each algebra topic.

So, what is algebra? Algebra is the generalisation of number. OK, maybe that's a bit showy offy! Algebra is maths with letters in it. That'll do for now!

Substitution and formulae is an area of algebra that involves a lot of number work so it's a perfect place to start our journey. It also gives an insight into how useful algebra can be and this is really important.

What's a formula?

Well, imagine you take a temperature reading on an old thermometer of 80℉ but because you live in the 21st century, you want the reading in Celcius (℃). No problem, you can convert from Fahrenheit to Celcius (roughly) as follows.

Take your Fahrenheit reading, subtract 32, multiply by 5 then divide by 9. This is what you do whenever you've got a Fahrenheit reading and you want to convert it to a Celcius reading. We call this a formula and it works whatever your Fahrenheit reading is.

This formula is far too wordy for mathematicians, though, so we write it using shorthand.

\(C=(F-32)\times \frac{5}{9}\)

And that's where the letters come from. In this case, C stands for the amount in Celcius and F stands for the amount in Fahrenheit. We don't have to use C and F, we could use any two letters so long as we make it clear which one represents the Celcius reading and which one represents the Fahrenheit reading. Lower-case letters are actually much more common in algebra. You must stick to whatever case is used in the question.

Because the value of these letters can vary (F can be anything you want it to be and C will vary accordingly), we call them variables. So if I talk about variables in algebra, I just mean letters.

A formula is made up of a variable representing the thing we want to work out (called the subject of the formula), followed by an equals sign, followed by an expression. An expression is just a mix of numbers, variables and mathematical operations (like \(~+~\), \(~-~\), \(~\times~\) and \(~\div~\)).

Formulae are incredibly useful and they're everywhere (geometry, science and engineering, computing, economics, business and finance, etc.).

What is substitution?

Let's use our formula to convert 80℉ to Celcius.

Well, \(~F=80~\) so we can put 80 in the formula in place of \(~F~\).

Putting a number in place of a variable like this is called substitution.

\(C=(80-32)\times \frac{5}{9}\)

If we work this out using a calculator, we get roughly 27, so 80℉ is roughly equivalent to 27℃.

Over the next few lessons, you're going to practise replacing variables with numbers (substitution) and evaluating the result. But you won't be working with real formulae until you get to the final lesson. Until then you'll be working with completely made up expressions! Making up expressions allows me to target specific skills and build these step by step. These skills will prove invaluable throughout algebra, geometry and many areas of science.

Practise to master

All lessons on this site are followed by lots of carefully thought out practice questions with fully worked solutions. This lesson is an exception since it is intended simply as a general introduction to the topic.

Before moving on, though, you should know what substitution means, what a variable is and what a mathematical operation is. You should also have an idea of what a formula is and you should kind of get what an expression is. Your understanding of these will improve as you work your way through the topic.