Simultaneous Equations

Today we are going to define and solve simple simultaneous equations. When presented with an algebra problem, you need to recognize which of them are simultaneous equations so you can solve them. Solving simultaneous equations is easy when you know what to do.

Definition of simultaneous equations

Simultaneous equations are two or more algebraic equations that are true at the same time. For example if you have two simultaneous equation:

Both of them must hold true. So, how do we find x and y given the above equations? First of all, there are a few ways of solving simultaneous equations, and what we will show you is one of them.

The first thing to do is take one of the equations. It doesn't matter which one - just take single one of them out to start from. Let's say, we choose x + y = 10. Now write the equation you just chose down.

Next we are going to use the second equation to substitute for either x or y. You may need to rearrange the equation before you substitute. So, using the second equation, we know that x = 4y so we can substitute x in the first equation.

You will get:

Now you have one equation with just one variable y, not y and x which you can easily solve. By adding 4y to y, you get 5y on the left hand side and 10 remains on the right hand side. That gives y = 2 as your solution for y.

The last step in solving this simultaneous equation is finding the solution for x. We know that x = 4y and now we know that y = 2. That gives x = 4 times 2 = 8.

That's it. Now you have solved the simultaneous equation above and found that x = 8 and y = 2. You can use this substitution method for any simultaneous equations.