Online College Courses for Credit

+
3 Tutorials that teach Solving a System of Linear Equations using the Addition Method
Take your pick:
Solving a System of Linear Equations using the Addition Method

Solving a System of Linear Equations using the Addition Method

Author: Sophia Tutorial
Description:

Solve a system of linear equations by using the Addition Method.

(more)
See More

Try Our College Algebra Course. For FREE.

Sophia’s self-paced online courses are a great way to save time and money as you earn credits eligible for transfer to many different colleges and universities.*

Begin Free Trial
No credit card required

47 Sophia partners guarantee credit transfer.

299 Institutions have accepted or given pre-approval for credit transfer.

* The American Council on Education's College Credit Recommendation Service (ACE Credit®) has evaluated and recommended college credit for 33 of Sophia’s online courses. Many different colleges and universities consider ACE CREDIT recommendations in determining the applicability to their course and degree programs.

Tutorial
what's covered
  1. Solutions to a System of Equations
  2. The Addition Method
  3. Multiplying by -1
  4. Multiplying by a Scalar Value

1. Solutions to a System of Equations

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.


2. The Addition 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.

term to know
Addition Method
also called the elimination method, a strategy to solving a system of equations by adding equations in order to cancel variable terms.

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:

table attributes columnalign left end attributes row cell 2 x minus 3 y equals 9 end cell row cell 5 x plus 3 y equals 5 end cell end table

Our system of equations
7 x equals 14
Add equations, y-terms cancel
x equals 2
Divide by 7

We noticed that we had opposite y-terms 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 x-terms summed to 7x, and our constant terms summed to 14, but what happened to the y-terms? 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 x-coordinate, we need to find the associated y-coordinate. 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:

2 x minus 3 y equals 9
An equation in the system
2 open parentheses 2 close parentheses minus 3 y equals 9
x=2
4 minus 3 y equals 9
Evaluate 2(2)
short dash 3 y equals 5
Subtract 4 from both sides
y equals short dash 5 over 3

Our solution for y

The solution to the system is: left parenthesis 2 comma space minus 5 over 3 right parenthesis.

big idea
Use the addition method when you notice like-terms with opposite coefficients between two equations. When you add the equations, the variable term will disappear from the equation, because the opposite coefficients canceled each other out.


3. Multiplying by -1

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:

table attributes columnalign left end attributes row cell 3 x space minus space 4 y space equals space 12 end cell row cell 3 x space plus space 8 y space equals space 21 end cell end table

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 x-terms 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 x-variable! Let's take one of the equations, and multiply the entire equation by –1:

3 x plus 8 y equals 21
An equation in the system
short dash 1 open parentheses 3 x plus 8 y equals 21 close parentheses
Multiply the entire equation by short dash 1
short dash 3 x minus 8 y minus short dash 21
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:

table attributes columnalign left end attributes row cell 3 x minus 4 y equals 12 end cell row cell short dash 3 x minus 8 y minus short dash 21 end cell end table

Our system of equations
short dash 12 y equals short dash 9
Add the two equations
y equals fraction numerator short dash 9 over denominator short dash 12 end fraction

Divide by short dash 12
y equals 0.75
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 back-substitution:

3 x minus 4 y equals 12
An equation in our system
3 x minus 4 open parentheses 0.75 close parentheses equals 12
y equals 0.75
3 x minus 3 equals 12
Evaluate 4(0.75)
3 x equals 15
Add 3 to both sides
x equals 5
Our solution for x

The solution to the system is: left parenthesis 5 comma space 0.75 right parenthesis .

big idea
Multiply an equation by –1 when you notice two identical terms between two equations. Multiplying by –1 will turn one of the identical terms into a like-term with opposite coefficients, allowing you to add the equations to cancel a variable term.


4. Multiplying by a Scalar Value

Let's consider the following system of equations

table attributes columnalign left end attributes row cell 2 x plus 3 y equals 8 end cell row cell short dash x minus 2 y equals short dash 3 end cell end table

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 x-terms 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:

short dash x minus 2 y equals short dash 3
An equation in our system
2 open parentheses short dash x minus 2 y equals short dash 3 close parentheses
Multiply the entire equation by 2
short dash 2 x minus 4 y equals short dash 6
An equivalent equation

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

table attributes columnalign left end attributes row cell 2 x plus 3 y equals 8 end cell row cell short dash 2 x minus 4 y equals short dash 6 end cell end table

Our new system of equations
short dash y equals 2
Add the equations; the x-terms cancel
y equals short dash 2
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.

2 x plus 3 y equals 8
An equation in our system
2 x plus 3 open parentheses short dash 2 close parentheses equals 8
Plug in -2 for y
2 x minus 6 equals 8
Evaluate 3(-2)
2 x equals 14
Add 6 to both sides
x equals 7
Divide by 2


Our solution for x

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

big idea
Multiply an equation by a scalar value when you notice that a coefficient of a variable term is a multiple of its like-term in another equation. You can even multiply by negative scalar values in order to create equations with opposite like-terms. Doing so will allow you to add the equations in order to eliminate a variable term.

summary
The solutions to a system of equations is a specific coordinate pair (x,y) that satisfies all equations in the system. The addition method, also called the elimination method, involves adding equations together to cancel out or eliminate one of the variables. We may need to multiply one or both of the equations by a number so that when the equations are added, one of the variables will be eliminated. Multiplying by -1 or multiplying by a scalar value may be necessary with the addition method.

Terms to Know
Addition Method

also called the elimination method, a strategy to solving a system of equations by adding equations in order to cancel variable terms