Introduction to Algebra

Introduction to Algebra | By Michael Milford

The first and biggest thing to understand is what a variable is in algebra. Say I have the following equation:

2 + 7 = 9

This is a really simple equation – everyone knows that “two plus seven equals nine”. This equation has three numbers in it – a ‘2’, a ‘7’ and a ‘9’. It also has two operations – a ‘+’ sign and an ‘=’ sign.

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Now, instead of one of these numbers, let’s write a letter in its place:

2 + 7 = 9
x + 7 = 9

Notice how now, instead of the number ‘2’, there is a letter ‘x’. If you read this new equation out aloud, you should say something like: “x plus seven equals nine”. Notice that instead of saying ‘two’ we now say ‘x’. ‘x’ is what we call a variable, or pronumeral. Variables are used to represent numbers.

So in this case, we know that the ‘x’ represents the number ‘2’, since we replaced the ‘2’ with the ‘x’. So if we wanted to write the equation again, but this time writing it with the number that ‘x’ is representing, we’d write a ‘2’ where the ‘x’ is now:

x + 7 = 9
2 + 7 = 9

Notice how we’ve ended up with our original, all number equation.

In this last example we used a letter (‘x’) as a variable. Variables can also be symbols. People sometimes use Greek Symbols as variables, such as θ, called ‘theta’. If we used θ instead of ‘x’ in the last equation, we’d write:

θ + 7 = 9

This time the ‘2’ would be represented by θ.
Now, what happens if we start with an equation which already has a variable in it:

5 – x = 2

This equation has the variable ‘x’ in it. This time though, we don’t already know what number the ‘x’ is representing. If we want to find out what ‘x’ represents, we need to “solve the equation for x”. This means – work out what value the ‘x’ is representing.

To solve the equation for x, we need to work out what number we can replace the ‘x’ with to make the equation true. So, we might as well try a number – let’s try ‘1’:

5 – 1 = 2?
4 does not equal 2

So instead of writing ‘x’, I wrote ‘1’ down, and then calculated what 5 – 1 equals – it equals 4. We are trying to get 2 as our answer, not 4, so using 1 does not make the equation true.

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Let’s try replacing ‘x’ with ‘2’:

5 – 2 = 2
3 does not equal 2

So using ‘2’ instead of ‘x’ doesn’t make the equation true either. When we put ‘2’ instead of ‘x’, we get an answer of 3, and we want an answer of 2. But we’re on the right track, cause 3 is only just bigger than 2. Let’s try replacing ‘x’ with ‘3’:

5 – 3 = 2
2 does equal 2

Bingo! When we use ‘3’ to replace ‘x’, we get an answer of ‘2’, which is what we want. We can say something like: “When x equals three, the equation is true.” If you were writing an answer to the question, “Solve the equation for x,” you’d write:

x = 3

Michael Milford, PhD

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