A solution to a system of linear equations is a specific coordinate pair (x, y) that satisfies all equations in the system. It is important that the solution satisfies every equation in the system, not just one; otherwise it doesn't represent a solution to the entire system.
There are several ways to find solutions to a system of linear equations, such as by graphing or using the substitution method. This lesson focuses on one method known as the addition method or the elimination method.
The main goal of the addition method is to add the equations that make up the system so that one of the variable terms will cancel, resulting in an equation with only one variable. From there, we can solve for that variable, and use that value to substitute it back into other equations in the system and eventually solve for the remaining variable(s). This is why this method is also referred to as the elimination method; because the goal is to eliminate one of the variable terms through equation addition.
In order for a term to cancel when we add equations, we just have a variable term in one equation, and its opposite in another equation. For example, if we have a 3x in one equation and a –3x in another equation, we can add the two equations together, and there will be no x term. Let's take a look at a concrete example:

Our system of equations 


Add equations, yterms cancel  

Divide by 7 
We noticed that we had opposite yterms in our system. In one equation, we had a positive 3y, and in the other, we had a negative 3y. When adding the equations together, our xterms summed to 7x, and our constant terms summed to 14, but what happened to the yterms? They disappeared, because the sum of a quantity and its opposite is zero. This eliminated an entire variable from our equation, allowing us to easily solve for x.
This doesn't represent our full solution to the system, however. Now that we know that 2 is our xcoordinate, we need to find the associated ycoordinate. To do so, choose any equation in the system  it does not matter which one we choose. We'll substitute 2 in for x and solve for y. This is shown below:

An equation in the system  

x=2  

Evaluate 2(2)  

Subtract 4 from both sides  

Our solution for y 
The solution to the system is: .
Using the Addition Method is ideal when we recognize a variable term in one equation, and its opposite in another equation. Unfortunately, this isn't always the case. Consider this system:
Adding the two equations as they are would not cancel any variable term. Sure, we notice that we have a –4 coefficient in front of y in one equation, and +8 coefficient in front of y in the other, but –4 and +8 are not opposites; they do not sum to zero.
Look at the xterms in each equation, however. Wouldn't it be nice if one of them were negative? That way, we would be able to add the two equations and eliminate the xvariable! Let's take one of the equations, and multiply the entire equation by –1:

An equation in the system  

Multiply the entire equation by  

An equivalent equation 
When we multiplied by –1, every term throughout the entire equation changed sign: positive terms became negative, and negative terms became positive. As a result, we have two terms that are opposites of each other, and we can use the addition method to eliminate a variable term. This is shown below:

Our system of equations 


Add the two equations  

Divide by 


Our solution for y 
Now that we have a value for y, we can once again use any equation in our system to solve for x through backsubstitution:

An equation in our system  



Evaluate 4(0.75)  

Add 3 to both sides  

Our solution for x 
The solution to the system is: .
Let's consider the following system of equations
Recall that in the above example, we noted –4y and 8y would not eliminate y, even though one term was positive and one term was negative. Looking at this example, what would make the xterms eliminate through addition? If we could turn –x into –2x, or if we could turn 2x into x, we would be in business. We can do just this by multiplying one of the equations by a scalar value. Let's take our second equation, and multiply it by 2:

An equation in our system  

Multiply the entire equation by 2  

An equivalent equation 
We can use this equivalent equation and add it to the other equation in our system to eliminate the xterms altogether, allowing us to solve for y.

Our new system of equations 


Add the equations; the xterms cancel  

Our solution for y 
With a solution for y, we can plug 2 in for y in any of the other equations in the system and find our solution for x.

An equation in our system  

Plug in 2 for y  

Evaluate 3(2)  

Add 6 to both sides  

Divide by 2  


Our solution for x 
The solution for the system of equations will be (7, 2).