Before You Watch

This video follows directly on from Linear Equations. It shows how to take two linear equations and solve them together to get a single value for the two variables (which are usually x and y, but could be any two letters). When we think about this process graphically, we are finding the point where the two lines intercept. This is called solving simultaneous equations. Keep in mind that in Australia this topic is known as simultaneous equations, but in other countries it is often referred to as “solving a system of equations”.

If you haven't watched Linear Equations yet, take a look at that video first, then come back.

The Video

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Some Practice Questions


Now What?

This video showed the simplest way of solving two linear equations simultaneously. In general, any two equations with the same two unknowns can be solved simultaneously. Bear in mind, however, that depending on the equations, that’s sometimes not possible. Let's think about it graphically: each equation is a line, and if the two lines cross, you can solve them. If they don’t cross, you won’t be able to find a real solution.

Moving beyond two equations, what if there are three, or four equations? If you have more equations, you can find more unknowns.

Once you are comfortable with solving two equations simultaneously, consider giving three equations a go at:

You should also consider having a look at how to solve other equations, such as Quadratic Equations or equations with Exponentials.

But When I am Going to Use This

Linear equations are very commonly used in everyday life to model situations. So when two things are both modelled by a linear equation, we can solve them together (simultaneously) to find out important information.

A common example of this is in finance and the analysis of revenue and costs. The revenue for a company from a product (say R) is simply the price of the product (let's say $30) multiplied by how many they sell (n), so

R = 30n

This relationship can be represented by a straight line. Similarly, the cost of that product (C) is a fixed starting amount (the development cost, staff wages and so on) plus whatever the cost of making the product is multiplied by how many they sell.

Let's say there is a fixed monthly cost of $1500 and the product costs $10 to make. That means the costs per month are:

C = 1500 + 10n

This can also be represented as a straight line. So when revenue is equal to costs (R = C), the company is breaking even. By solving these two equations simultaneously:

C = 30n

C = 1500 + 10n

we can determine how many products, n, must be sold per month for the company to break even. See if you can work it out, the answer is n = 75. A graph of the functions is given below.


This is just a single example. Some other examples can be found at: .

Other Links

She Loves Maths has a wide range of pages covering many mathematical topics across all levels. The pages cover the topic using practical, relatable examples. This content deals with systems of linear equations.

Maths is Fun provides a clear summary of solving equations and explains the different terminologies used. It also offers several questions to practise on.

The Khan Academy has a comprehensive set of video tutorials covering a large range of mathematical and other concepts, as well as questions to test your knowledge. This link is to a chapter dedicated to solving equations, with over a dozen videos and several quizzes.

Patrick JMT (Just Maths Tutorials) has many video tutorials covering a large range of mathematical concepts. The content below demonstrates three different techniques for solving a system of linear equations. Any of these methods can be used to solve any system of equations but, depending on the exact question, one strategy may be easier to use than the others.