This is for all of those parents out there who are trying to help their children to understand their Maths homework, but are struggling to understand it themselves! I mean, we all know that 2+2=4 and many of us will know our times tables and even know how to divide, but what the heck does 2+b equal?? As Billy Conelly once said, “I don’t think I was in for the b times table!” I will try to demystify algebra with some of the basic tools to start to help make up an algebra tool kit.
I shall start by explaining some terminology that is useful for algebra and which is usually expected to be learnt by students. If any of this seems confusing for now, don’t worry, just read on and it might be less confusing as you continue.
- variable: This is the unknown value in the equation represented by a letter, such as x.
- operator: This refers to the symbol for adding, subtracting, multiplying, dividing and so forth.
- equation: This is like a statement using variables, numbers and operators to say something.
Eg. x + 3 = 5
This means “If I add the variable x together with 3, it would make 5.”
Equations must have equal signs otherwise it would not be saying anything.
- expression: These are often confused with equations, but do not have equal signs. They don’t tell you any information. x+5 is an expression, for example.
- product: This what you get when you multiply two values together. The product of 3 and 5 is 15. When you multiply a variable by a number, you can show the product by writing the number to the left of the variable. Eg. 3a means 3 multiplied by a. Two variables multiplied together are written next to each other in any order. Eg. xy or yx.
- term: This is anything that is being added or subtracted to something else within an expression. This could include products.
- solve: This means to find the value of the variable. It is not always possible to do this.
- rearrange: This means to change the equation in some way, which may help to use it more easily, and may help to solve it.
Solving an equation
Let’s start by looking at the equation below:
2x + 5 = 11
x, or whatever letter is there, represents an unknown value, or a variable. We call it a variable because the value of it could vary. If the equation has more than one variable it cannot (usually) be solved. If a number is written to the left of a variable, like ‘2x’ as above, it means 2 multiplied by x. This little fact is often not understood by some students early on and can cause huge confusion later on.
Think of x as a box of apples, labelled x, which cannot be opened. All of these x-boxes will have the same number of apples in. You can now tell your son or daughter that you’ve been learning about x-boxes! What the equation is saying is:
“Two boxes and 5 apples will add up to 11 apples”
Thinking of it like this, you might realise that 2 boxes and 4 apples would add up to 10 apples. In fact two boxes and no apples would add up to 6 apples. I’ve just taken 5 apples from the left and 5 from the right.
Remember that each x has the same number of apples, so it must be 3 each! I just divided both sides of the equation by 2.
My Golden rule with rearranging equations
What we did there was rearrange and solve an algebraic equation. There were two steps.
Step 1: Take away 5 from both sides (of the equals sign).
Step 2: Divide both sides by 2.
Probably the most important thing to remember when rearranging equations is:
Whatever you do to one side, you must do the same thing to the other side.
If we were to solve the above equation without using my (hopefully) helpful boxes of apples analogy, it would look like this:
2x + 5 = 11
Step 1: Subtract 5 from both sides.
2x + 5 – 5 = 11 – 5
Adding 5 and then subtracting cancels out, so…
2x = 6
Step 2: divide both sides by 2
x = 3
When deciding on what step to take, you are trying to free the variable, so as it is finally on its on own on one side of the equals sign, with a number on the other side. To free it, you should ask “What is currently being done to the side it is on?” At first 5 was being added to the left, so I did the opposite. I subtracted 5. Next, x was being multiplied by 2, so I did the opposite. I divided both sides by 2.
This is, more or less, how you do algebra. Well… it should get you started anyway! Good luck!
A bit about the author, Paul H:
Paul is a qualified and experienced Physics, Maths, and Science teacher, now working as a full-time tutor, providing online tuition using a variety of hi-tech resources to provide engaging and interesting lessons. He covers Physics, Chemistry, Biology, and Science from Prep and Key Stage 3 through to GCSE and IGCSE. He also teaches Physics, Maths, and Chemistry to A-Level across all the major Exam Boards.