Have you ever wondered while studying algebra ... "will this information EVER be used?!?” Well that is what I thought until I learnt some applications of mathematics later on in my study.

Did you know that the traffic lights at the end of YOUR road are actually controlled by algebra?

Hmmm... How does this work?

Let’s say that there is a traffic light at the end of your road and the time the traffic light is red is represented by x.

Now if the traffic light can only be red at any one time for say 2 minutes, we could represent this information in an algebraic equation:

x = 2    that is x (the time, in minutes, the traffic light is red) is equal to 2 minutes.

Now, what if there were two sets of traffic lights at a junction. The first set of traffic lights can only be red for 2 minutes and the second set can only be red for 4 minutes.

Using the above definition, we could interpret this information as the following algebraic equations:

x = 2    => first set of traffic lights can only be red for 2 minutes

y = 4    => second set of traffic lights can only be red for 4 minutes

These are quite simple definitions.

Let’s make it more complex. What if we did not know for how long each traffic light could be red but we knew that the sum of this time had to equal 10 minutes.

Again, we could represent this as an algebraic equation where

x = time (in mins) traffic light 1 is red

y = time (in mins) traffic light 2 is red

x + y = 10          =>   time traffic light 1 is red + time traffic light 2 is red = 10 mins

So we have that the total time both traffic lights are red is 10 minutes.

What if we had a further restriction such that the difference between the times the traffic lights are red is 6 minutes.

We can represent these two restrictions below (keep in mind the definition for x and y):

x + y = 10

x – y = 6

You will have come across a set of algebraic equations known as ‘Simultaneous Equations’. The two equations above can also be called Simultaneous Equations as we are looking for a value for x and y such that both equations are true.

Give it a go!

Hopefully you should end up with x = 8 and y = 2. Translating this into simple terms, traffic light 1 should be red for 8 minutes and traffic light 2 should be red for 2 minutes for the above equations to be simultaneously true ... I feel a bit sorry for the cars travelling through traffic light 1.

Well the above was quite a simple explanation. In reality, sorting traffic lights at a junction will contain many constraints with one main objective function. The objective will be to minimise waiting time for cars at the junction and this objective will be subjected to constraints, like the ones we have mentioned above but perhaps with more complexity such as taking into account crossing times for pedestrians, any extra time for traffic specifically turning right or left, ensuring that both traffic lights are not red at the same time, both traffic lights are not green at the same time and other factors like one traffic light having a lighter flow of traffic than the other so it can remain red for longer.

The whole junction problem can be represented as a set of algebraic equations or expressions (lots of them) which are then solved simultaneously using a computer program. This concept is part of mathematical modelling (in particular, linear/quadratic programming) and its applications are worldwide such as maximising profits/sales, minimising costs/overheads, say, for airline companies and portfolio investment. Other areas include minimising time wastage on a production line, allocating limited resources/human resources, distribution, marketing etc.

Maybe algebra won’t seem so boring anymore?