Simultaneous Equations

Solving simultaneous equations usually involves two linear equations with x and y unknowns, and the goal is to find the value of the unknowns which satisfy both equations. The process derives the solutions to both equations simultaneously, and the values of x and y, when substituted within these equations, satisfy both equations.

There are many ways to solve these equations, for example, you could draw graphs of the two linear equations, and look for the point of intersection. Where the two lines cross, is the point where the values of x and y is the same for both equations. The other method involves algebraic manipulation of the two equations, and this article looks into how that works.

The algebraic method is extremely simple, however when the “experts” do it, they sometimes add the two equations to eliminate one unknown, and sometimes they subtract the two equations, and if you are just starting out, you might find it confusing. Therefore, I will show one system and process that works for all equations.

On-line Solver

The Simultaneous Equations Solver is an extremely powerful online calculator that will do your homework for you. If you are a teacher, then it will also serve as a tool for making interesting questions.

People are always looking for an app for the TI-84calculator to do all the work for them, consequently students end up with A grades but know very little about solving these by hand. My advice would be to learn how to solve these by hand. Certainly if you want to become a pilot or work for the RAF then you are going to need to know this.

Back in my day, we had what would be now considered Vintage Calculators, however the good news is that modern calculators can give you the answers, show you the working out, and draw a graph!

Rules

The best way to learn is by going through as many examples as you can to understand the principle. You learn by doing, and the more you do, the more experience you will gain. It is actually a very simple thing but for some reason the language required to describe the steps always makes it appear more complicated than it really is.

Elimination

There are two main steps to solving these types of equations. Moreover, it always begins with the elimination step. The goal is to eliminate one of the two unknowns. It could be either x or y. It does not matter which you eliminate first.

The elimination step always involves writing one equation on top of the other and aligning the like coefficients. The idea is to subtract the like coefficients in a manner, which eliminates one of the unknowns.

Substitution

The second step is always the substitution step. Once you have found the value of either, x or y, then the logical thing to do would be to substitute it into any of the previous equations to find the remaining unknown. What else can you do?

Simultaneous equations can be confusing for beginners. Especially when your teacher goes through five different examples and in each, they do everything differently. In one example they might divide throughout, then in another they might multiply throughout. Therefore, when you have to solve it you do not have a clue. What are you supposed to do, should you multiply, or divide, or subtract, or add, or what?

Divide, Subtract, Add, or What?

If you are new to solving these equations then you will not have the experience to spot the short cuts, therefore, it is probably better to start by using one hard and fast rule that works for all the equations. Then later perhaps when you get more experienced you can look for short cuts.

Two Linear Equations

In the hard and fast rule, you only ever multiply throughout, and you do everything the same way. Here are a pair of linear equations where a, b, c, d, e, and f are numbers. The goal here is to eliminate the first set of terms in both equations, thereby eliminating x. We then have only the y term in both equations, which we can solve easily.

Elimination Method

Multiply the top equation throughout by d, and multiply the bottom equation throughout by a, which gives you the result shown above. Since you are multiplying throughout, the equations remain balanced.

Eliminate x term

As you can see, the first terms in both equations are the same and positive. You can therefore subtract them, which eliminates them including x.

It does not matter which way round you subtract the equations as long as you pay attention to your negative signs. Nevertheless, for simplicity and convention we are subtraction the bottom equation from the top since both are positive.

x cancels out.

As you can see subtracting the first set of terms will eliminate x, because our multiplication made them the same. You can use this method of elimination for any pair of equations where the first terms are positive.

If the one of the terms was positive and another negative, then you could add them, which will cancel them out. If you are adding, then remember that you also then have to add the second set of terms, and the third set of terms.

In this example, we are subtracting because the first set of terms, which we want to eliminate, are positive.

You then subtract the second set of terms, which contains the y, and the last set of terms after the equal sign as shown above. I have factorised out the y to make it simpler.

Solution for y

Then it is a simple matter of solving for y. When you have numbers instead of letters, it is even simpler. However, what I have done here is to derive the formula to find y, where a, b, c, d, e, and f are numbers belonging to a pair of linear equations, and a and d are positive.

Once you have a value for y, you just substitute it into any one of the equations to find x. It does not matter which of the two linear equations you use, you can substitute y into either to find x.

Worksheet Example Questions

Here are some practice questions and answers that might prove useful. For those more advanced you can treat them as a quiz. I have made these to the same style as GCSE, which might be useful for teachers as well

GCSE Style Question and Answer 1
GCSE Style Question and Answer 2
GCSE Style Question and Answer 3

Homework Section

Equation 1 Equation 2
6x+2y=-3 4x-3y=11 6x+2y=-3 and 4x-3y=11
3x+2y=4 4x+3y=7 3x+2y=4 and 4x+3y=7
4x+y=-1 4x-3y=7 4x+y=-1 and 4x-3y=7
3x+2y=8 2x+5y=-2 3x+2y=8 and 2x+5y=-2
x+y=4 x^2+y^2=40 x+y=4 and x^2+y^2=40