In this section we will explain step by step the substitution method to solve systems of two equations with two unknowns. There are also other methods of resolution, such as equalizing and reducing, but I will focus only on the substitution method.
Basically, the substitution method consists of:
- Clear an unknown in one of the equations, which will depend on the other unknown (we will continue to have one equation).
- In the other equation that we have not used, the same unknown is replaced by the value obtained in step 1.
- Clear the only question we have left. We get the numerical value of an unknown.
- Replace the unknown value cleared in step 3 with its numerical value (also obtained in step 3) in the equation obtained in step 1.
- Operate to obtain the numerical value of the other unknown.
Let’s take a slower look at the substitution method with a step-by-step example.
Step-by-step Substitution Method
Let’s solve for example the following system of equations:
To know at all times which equation of the system we are referring to, the equation above will be called the first equation and the second equation below:
1- We clear an unknown in one of the equations, taking into account the rules of transposition of terms.
The easiest to clear is the «y» in the first equation, because it has no numbers in front of it and also has a sign in front of it, so just passing the 5x on the other side and we have the clear one:
This is at the moment our «y» value, which we say is based on x, because x is contained in your result. We also highlight it by locking it in a red box, because we will have to return to this equation later.
2- In the equation that we have not used, we replace the same unknown that was cleared in the previous step with the value we have obtained.
That is, in the second equation, where «y» appears, we replace it with its value as a function of x:
We have one equation left that only depends on an unknown.
3 – We clear the unknown we have left.
Now we have an equation that depends only on «x».
We solve the equation we have left:
And we get the numerical value of x.
4 – The numerical value obtained is replaced in the equation where we clear one unknown as a function of another (step 1). In our case, where we’re clear and based on x:
We replace x with its value:
5 – And we operate to obtain the numerical value of the unknown we have left:
Therefore, the solution of this system is x=2, y=-2.
Substitution method: When it should be used
The substitution method is the most widely used of the three, as it is the most versatile. Use it if you are not told otherwise in your exercises.
So, to the question of when do I have to use the substitution method? The answer is that when you are not clear about which method to use, use the substitution method. 95% of cases.