Your Ultimate Guide to Solving Simultaneous Equations

Unsure of how to solve simultaneous equations? No worries. Read on!! 

Solving simultaneous equations is a fundamental skill that will help you in both Junior and Senior Maths. Since the applications of simultaneous equations get harder as you progress through the later years of study, knowing the basics of simultaneous equations is key to performing well in later topics. 

In this post, we’ll start by introducing what ‘simultaneous equations’ are and how we approach solving simultaneous equations generally. We’ll then explore the two methods of solving simultaneous equations: Elimination and substitution.

Let’s dive right in!! 

1. What are Simultaneous Equations?

With simultaneous equations, we are given a pair of equations with two unknown variables. An example is as follows:

2x-y=4

3x+y=5

In this case, our two unknown variables are x and y. This is distinct from simple linear equations (e.g. 2x-3=5) or quadratic equations (e.g. x2-5x+6=0) with only one unknown variable (x). 

When solving ‘simultaneous equations’, the ‘general approach’ we need to take is to get rid of one of the pronumerals. We can do that by using elimination or substitution. After we have done this and found the value of one of the pronumerals, we can then go back and find the value of the other pronumeral. 

That being said, let’s now go through our two methods of solving simultaneous equations.

2. Methods of Solving Simultaneous Equations

2.1 Method #1: Elimination

Let’s say we wanted to solve these two equations:

2x-y=4    – (1)

3x+y=5    -(2)

With elimination, what we are trying to do is add or subtract the equations in a way that will eliminate one of the pronumerals. In this case, let’s try to eliminate the y variable. 

To do this, let’s add the two equations from each other, remembering that we add the left hand side terms from equation (1) by the left hand side terms from equation (2) (and same for the terms on the right hand side) :

As we can see, we’ve eliminated the y variable. Now solve for x:

5x=9

x=95

Now that we have x, all that is left for us to do is substitute it back into one of the equations to find y. Let’s choose equation (2) (although you’ll still get the same answer for y if you use equation (1)):

3(95)+y=5

y=5-275

y=-25

So our answers are x=95; y=-25.

Now, here are a few other things to note.

Firstly, be mindful of the sign of the pronumerals. In the example above, let’s say that we had 3x-y=5 for the second equation as opposed to 3x+y=5. Here, we would need to subtract the equations rather than add them to get rid of y.

Secondly, you might have some instances where elimination is a bit more tricky. For example, take the equations 3x+5y=1 and 5x+2y=4. In this situation, what we’d do is choose our pronumeral and find the lowest common multiple of the coefficients before the pronumeral we have chosen. So, let’s choose y again:

3x+5y=1   -(1)

5x+2y=4   -(2)

Looking at 5y and 2y, we can see that the lowest common multiple of 5 and 2 is 10. To get 10y in both equations, we’d need to do the following:

2(3x+5y)=12   -(1)

6x+10y=2   -(1)

And:

5(5x+2y)=45 -(2)

25x+10y=20    -(2)

Now we can proceed with elimination, considering the new adapted equations:

And then we’d solve for x and y.

2.2 Method #2: Substitution

Let’s consider the following equations:

2x-y=13

y=3x+6

With substitution, what we are doing is ‘substituting out’ one of the pronumerals and replacing it with an algebraic expression containing the other pronumeral. 

Consider this:

2x-y=13    -(1)

y=3x+6    -(2)

Substituting equation (2) into equation (1) for y:

2x-(3x+6)=13

And solve:

2x-3x-6=13

-x=19

x=-19

Then we’d go back and substitute the value for x back into one of the equations to get y. Let’s pick equation (2):

y=3(-19)+6

y=-51

So x=-19; y=-51.

2.3 When do I Know Which Method I Need to Use?

Some situations will lend themselves towards one method over the other. For example, if you have a situation where one of the equations already has one of the pronumerals as the subject of the equation, then you’d use substitution. On the other hand, if you have two equations that do not have one of the pronumerals as the subject of the equation, you’d consider using elimination.

Now, this is not to say that you can’t use substitution for a question that would lend itself towards elimination. You can always rearrange one of the equations to make a particular pronumeral the subject and solve from there. It’s up to you!!

3. Final Thoughts

So there we have it, our guide to solving simultaneous equations. To recap:

• Solving simultaneous equations requires us to start by getting rid of one of the pronumerals. After we’ve found the value of one of the pronumerals, we then go back to solve for the other. 
• We can do this by using two methods: Elimination and substitution.
• Elimination involves adding or subtracting the equations to ‘eliminate’ one of the pronumerals.
• Substitution involves ‘substituting out’ one of the pronumerals for its equivalent value in terms of the other pronumeral. 

From here, the best course of action is to spend time reading and re-reading this blog article and to practise both methods. This way, you’ll be well-equipped to deal with whatever questions you may encounter.

Good luck!!