Algebra: The Great Mathematical Whodunnit

A few weeks ago I put a little bit of algebra in our weekly quiz on Instagram Stories. Nothing too heavy, just a few linear equations to test the water. Well, it turned out the water was fine and you all jumped in feet first. In fact, raving algebra addicts that you apparently are, a whopping 92% of you voted for a second week of algebra. And then you demanded a third week so apparently we’re now building an algebra army and I can’t express just how okay I am with that. Here’s a quick recap of everything we’ve learnt so far. The word ‘algebra’ comes from the Arabic ‘al-jabr’ which means ‘bone setting’. We use it in the sense of restoring something that’s broken. The thing that’s broken is an equation. Okay. What’s that?

An equation is just a statement that says that two things are equal. 3+2=5 is technically an equation. We’ve got two things either side of an equals sign and we know that those things are equal.

 But imagine if a bit of the equation got broken off and went missing. Ciao for now number 3.

Before poor number 3’s bed was even cold, the international letter of mystery, x, would jump right in to take his place and we’d have something like this:

This ‘x’ is called an unknown. We don’t know what it is… yet. It’s our job as good bonesetters to figure out what’s missing, to restore our broken equation, to make it complete again.

 Now, your inner Sherlock Holmes might be jumping out of his tweed cape yelling, “It’s 3!” and he’d be absolutely right so give him a biscuit but tell him to sit down for a minute because the Case of the Missing Number won’t always be so easy to crack, so we need a method that will work even when the answer isn’t quite so… obvious. No offence Sherlock. That method is going to be ‘balancing’.

We can think of an equation like a set of weighing scales where the two sides are completely balanced, and we have one job: to keep it balanced. It's a bit like a very low stakes Die Hard 3. If we add something to one side, we have to add that thing to the other side. If we take something away from one side, we have to take it away from the other side. Them’s the rules. Now let’s see how to play.

In an equation like x+2=5, we want to know what the x is. If we can get it on its own on one side of the equals sign, we’ll know exactly what it is because it’ll be equal to whatever’s on the other side of the equals sign. To get it on its own, we have to undo anything that’s been done to it, by doing the opposite, like a reverse spell.

Our x has had a 2 added to it so the reverse spell would be to subtract a 2, that’s the opposite of adding a 2. If we started with an x then added 2 then took 2 away again, we’d end up with just x, which is exactly what we want. But remember, we’ve got to keep our scales balanced. If we’re going to subtract a 2 from one side, we have to subtract a 2 from the other side. Now we’ve got something that looks like this:

If we clean this up a bit we’ll end up with x on the left hand side and 3 on the right hand side, so x=3, exactly as Sherlock suspected. We can check our answer by putting the 3 in place of the x in our original equation (x+2=5) and we’ll see that 3+2 does in fact equal 5. Case cracked.

Let’s have a look at another example.

This time we’ve got x-3=4. Once again, our quest is to get the x on its own by undoing whatever’s been done to it with a reverse spell, keeping our equation balanced at all times and looking good while we do it. Move over John McClane, we’ve got this one.

This x has had a 3 taken away from it. The reverse spell is going to do the opposite, it’ll add a 3. But, if we add a 3 on one side, we have to add a 3 on the other side. Now we’ve got:

A bit of housekeeping leaves us with just an x on the left hand side, exactly as we planned it, and a 7 on the other side. x=7. For the grand finale, we stick that 7 in place of the x in our original equation (x-3=4) to check that everything works. And it does. Take a bow! 

Alright, the only thing stopping us from being the greatest (and only) algebra army the world has ever known, is a bit of practice. Have a go at these questions and remember: reverse the spell to get the x on its own, keep the equation balanced. Next time we’ll be reversing multiplication and division. X-cellent work everyone. Fall out.