Simultaneous equations are a group of equations that each use the same variables and, therefore, have the same solutions. You may be presented with two simultaneous equations or more in an exam. The information contained within each of those equations can be used to solve all of them.

Solving equations of all types is one of the most critical components of algebraic mathematics. But it is also one of the areas that many maths students most dread due to its renowned complexity. However, as we will see, there are several simple methods you can use to help you solve an equation.

So how do you solve simultaneous equations?

Join us as we explore the main methods for solving simultaneous equations.

There are several ways of solving simultaneous equations. The most common methods are the substitution method, the elimination method, the algebraic method, and the graph method.

We will take you through a step-by-step guide to each of these methods, but let's first find out some of the key terms and definitions that are useful for solving equations.

Terms and definitions

Here are some of the key terms and definitions used for simultaneous equations and equations in general:

  • Equation. In mathematics, an equation is a formula that expresses the equality of two expressions. Equation means equal. We can find the value of unknown variables by knowing that one expression equals another.
  • Variables. Also known as "unknowns," the variables in an equation are the unknown values often represented by letters a, b, x, and y.
  • Quadratic equation. Quadratic equations are equations in which the variables are squared.

What are simultaneous equations?

Simultaneous equations are two or more algebraic equations that have the same variables.

There are many types of simultaneous equations, but we are going to focus specifically on one type:

  • linear simultaneous equations. Linear simultaneous equations are equations in which there are no squared variables. Instead, they represent straight lines

Individual equations that have more than one unknown value have many different solutions. For example, 2x + y = 10 could be solved by:

  • x = 1 and y = 8
  • x = 2 and y = 6
  • x = 3 and y = 4

However, when you are presented with simultaneous equations, you can use the information from each of them to find the unknown values within them. Because the equations are solved at the same time, they are called simultaneous equations.

For example:

  • 2x + y = 12
  • 6x + 5y = 40

This is only true when the values of x and y are:

  • x = 5
  • y = 2

So let's now find out how we arrive at these answers.

How do you solve simultaneous equations using the substitution method?

The substitution method involves substituting the values from one equation into the other.

For example, let's say you are given the following simultaneous equations:

  • y = 2x
  • x + y = 6

The first step is to number the equations:

  1. y = 2x
  2. x + y = 6

The first equation tells us that the value of y is 2x. So substitute the value of y into the second equation, and you get:

  • x + 2x = 6

Because we now have three lots of x in the equation, it can also be expressed as:

  • 3x = 6

So now we know that x must equal two because:

  • 3 x 2 = 6
  • x = 2

We now have the value of x, but we still need the value of y.

Equation 1 informs us the y = 2x. Which means that y = 2 x 2.

Therefore, y = 4.

Now we need to check our answers by putting the values we have found into the original equations.

Equation 1 told us:

  • y = 2x
  • 4 = 2 x 2

Equation 1 works. So let's now check the other equation.

Equation 2 told us:

  • x + y = 6
  • 2 + 4 = 6

Equation 2 also works.

By using the substitution method, we have discovered that:

  • y = 4
  • x = 2

How do you solve simultaneous equations using the algebraic method?

The algebraic method of solving simultaneous equations involves adding or subtracting the equations to arrive at an equation with only one unknown value.

For example, let's say you are given the following two equations:

  1. 2x + y = 9
  2. 3x - y = 1

If we combine these two equations to get one big equation, we get:

  • 3x + 2x + y - y = 9 + 1

The y values cancel one another out, and the x values can be combined to make 5x. Therefore, this equation can be simplified to:

  • 5x = 10
  • 5 x 2 = 10

So we now know that:

  • x = 2

We can now substitute the value of x into our original equations to get:

  1. 4 + y = 9
  2. 6 - y = 1

The only number that works for these equations is 5. Therefore:

  • x = 2
  • y = 5

How do you solve simultaneous equations using the graph method?

The graph method allows us to solve simultaneous equations by plotting values of x and y onto a graph and finding the point of intersection. The point of intersection gives the solution to the equation.

For example, let's say the two equations we are solving are:

  1. y = x + 1
  2. x + y = 5

First, we need to create a table of values that offers potential solutions to each equation.

Here is a table of values for y = x + 1:


Here is a table of values for x + y = 5


Then, you need to draw a graph and plot these two tables of values onto the x and y axis to create two straight lines.

Find the point where the two lines intersect. The point of intersection on the x-axis is the solution for the value of x. The point of intersection on the y-axis is the solution for the value of y.

In this instance:

  • x = 2
  • y = 3

How do you solve simultaneous equations using the elimination method?

Another common method for solving simultaneous equations is the elimination method. The elimination method is similar to the substitution method, so let's look at it now.

Let's say you are given the equations:

  1. 3x + y = 11
  2. 2x + y = 8

You need first to identify which of the unknown values is equal in each equation; this is known as finding the 'coefficient.' In our example, the coefficient is y.

We then want to eliminate y from the two equations.

We know that y - y = 0. This is always true, whatever the value of y.

So we must subtract equation 2 from equation 1:

  • 3x - 2x = x
  • y - y = 0
  • 11 - 8 = 3

This means that we are left with:

  • x = 3

We can then substitute the value of x into our earlier equations:

  1. 9 + y = 11
  2. 6 + y = 8

Therefore, we know that:

  • y = 2

How do you solve simultaneous equations with no common coefficients?

As we have just seen, finding the coefficient is an important step in solving simultaneous equations using the elimination method. However, what if there is no common coefficient?

For example, let's say you are given the equations:

  1. 3x + 2y = 17
  2. 4x - y = 30

There are no common coefficients in these equations, as no values are equal.

So, we need to create a common coefficient by converting one of the equations.

The second equation has a single value of y, and the first has 2y. So we can multiply the entire second equation by 2 to create a common coefficient with the first equation:

  • 2(4x - y) = 8x - 2y

Our two equations are now:

  1. 3x + 2y = 17
  2. 8x - 2y = 60

We want our y values to cancel each other out and equal 0.

Look at the mathematical symbols before the common coefficients. If the signs are different, the equations should be added together. If the signs are the same, the equations should be subtracted.

Our symbols are different, so we will add the two equations together. This gives us:

  • 11x = 77
  • x = 7

Now we know the value of x, we can transfer it into the original equations:

  1. 21 + 2y = 17
  2. 28 - y = 30

We can see now that y must be -2. So our answers are:

  • x = 7
  • y = -2

There are four main methods of solving simultaneous equations. An exam question may specify which method you need to use. If it doesn't, then you should use whichever method you are most confident using or whichever method lends itself best to the question.

Students often dread equation-solving questions. But try using the step-by-step methods in our guide, and you will see that there is nothing to fear. You may even find that solving equations can be very satisfying and even (dare we say it) fun!