Elimination Method in Algebra: Definition & Examples

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  • 0:02 Systems of Linear Equations
  • 0:50 The Elimination Method
  • 2:55 Creating Opposites
  • 4:13 System With Infinite Solutions
  • 5:17 No Solution Means…
  • 5:57 Lesson Summary
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Lesson Transcript
Instructor: Ellen Manchester
There are three ways to find values of 'x' and 'y' when dealing with systems of two linear equations. In this lesson, we will be learning one of these methods: elimination.

Systems of Linear Equations

A system of linear equations is when there are two or three equations, with two or three variables. In this lesson, we will be working with a two-equation system, and we will be solving for both x and y. This answer, graphically, is where the two lines cross. If they do not cross, then we have parallel lines, and there will be no answer for x and y. If the equations are the same one, we have graphically, one line sitting on another, and we have an infinite number of solutions.

When dealing with systems of linear equations, there are three ways to solve for x and y: graphing, substitution, or elimination. In this lesson, we will be looking at the elimination method for solving a system of two linear equations.

The Elimination Method

The elimination method is where you actually eliminate one of the variables by adding the two equations. In this way, you eliminate one variable so you can solve for the other variable. In a two-equation system, since you have two variables, eliminating one makes the process of solving for the other quite easy. Let's try one:

Elimination Method Example 1

In the above example, you can see you have two equations with two variables. The goal is to have one variable that is positive and one negative in the two equations, so it is easy to eliminate. In this instance, we have -5y and 5y, so we can add the two equations, which eliminates the y-term. Once the y-term is eliminated, we use our basic Algebra skills to solve for the x-term. Once we add 5y to -5y, we can see that we have the equation 7x = 14. We divide both sides by 7 to isolate the x-term and get x = 2. Now that we have the x-term, we can use substitution to solve for the y-term. Plug 2 into either of the original equations to solve for y.

Elimination Method, substitution step

If we use the first term, we get 3(2) - 5y = -9 or 6 - 5y = -9. We subtract 6 from both sides and get -5y = -15. We divide both sides by -5 to get the value of y, which is 3. But remember, we can use either equation, so let's look at the other one to see if we get the same answer.

This time, we get 4(2) + 5y = 23 or 8 + 5y = 23. We subtract both sides by 8 and get 5y = 15. We divide both sides by 5 and get y = 3. As you can see, we got the same answer regardless of which equation we use. Let's look at the graph of these equations:

Graph of Example 1 Solution
Graph of Example 1

From the graph you can see the two lines intersect at the point (2,3). This is the solution for this system of equations.

Creating Opposites

Sometimes you will encounter a system of equations where you will need to create a system where you can eliminate a variable. This is done by multiplying values through the entire equation to set up one positive variable and one negative variable. Let's try this one:

Elimination Example 2

As you look at this example, think about which variable you want to eliminate. If we want to eliminate the x-term, we need to make sure we have a positive and a negative of the same term. I could multiply the first equation through by -3, which will allow me to eliminate the x-term.

Now we can eliminate the x-term because the -3x and 3x cancel each other out. We are left with y = -6. Then we can plug that number into the original equation and solve for x. As you can see, we have x - (-6) = 3, or x + 6 = 3. We subtract 6 from both sides and end up with x = -3. Therefore, the solution for this system of equations is (-3,-6).

Here is the graphic solution showing the two lines crossing at the point (-3,-6):

Graph of Example 2
Graph of Ex. 2

When two lines intersect, there is at least one solution. This is called a consistent system. A consistent solution with exactly one solution is called independent.

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