Maths for Diploma Studies

There are times when we may encounter an equation where we have two terms with unknown values. In this situation, you could be forgiven for concluding that there is no way to find the value for both. Fortunately, there is a method we can use to solve these problems. They are called simultaneous equations.

Equations that have two unknowns can seem to have an infinite number of solutions. For example in:

x could equal 1, and y could equal 8

or

x could equal 2, and y could equal 5

or

x could equal 3, and y could equal 2

And that’s not accounting for decimals!

So how do we find the values of the terms in the equation above? The approach is called simultaneous equations because we use a pair of equations to find the values. Here are the equations:

To use simultaneous equations, the first step is to get a common coefficient in the equations. A coefficient is a number multiplying a value. At the moment, we have a coefficient of 3 in the first equation and 2 in the second. We can get a common coefficient by multiplying these together like so:

And now we subtract one equation from the other:

Or:

Now we can input the value of 2 for y into the simplest equation from above:

And we can rearrange the equation to isolate x by subtracting 2 from both sides of the equation then dividing both sides by 2 and  :

And then doing the calculation:

We can use both values in our original equation to check it makes sense:

Job done!

We can use or simultaneous equation solving skills on real-life problems of critical importance. We want to revive an old recipe for grandmother’s famous cake, but the recipes we’ve found have missing elements and some odd units of measurement. Here’s what we know:

Don’t get distracted by the use of scoops and handfuls. We just need to treat them the same way we did the xs and ys earlier.

First, let’s get our common coefficient:

Now we can subtract one equation from the other:

Now we can rearrange this to find out the weight of a scoop of sugar:

And now we can substitute this value for sugar scoops into the simplest of our simultaneous equations:

And then rearrange the equation to isolate a handful of flour:

And calculate the value:

Finally, to check, we can use the values in the original equations:

We have the recipe. Let’s hope the cake was worth the effort!

To help reinforce what we have just learned, on the next page there are some practice questions. Answers and explanations can be revealed by clicking on show answer.