If you're seeing this message, it means we're having trouble loading external resources on our website. Show If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. The elimination method is one of the most widely used techniques for solving systems of equations.  Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher) Why? Because it enables us to eliminate or get rid of one of the variables, so we can solve a more simplified equation. Some textbooks refer to the elimination method as the addition method or the method of linear combination. This is because we are going to combine two equations with addition! Here’s how it works. First, we align each equation so that like variables are organized into columns. Second, we eliminate a variable.
Thirdly, we substitute this value back into one of the original equations and solve for the other variable. Simple! Solve the System of Equations by the Elimination Method
The reason is that addition is commutative, so no matter if we switch the order of the equations we will still arrive at an appropriate answer. Subtraction is not commutative, therefore could yield an incorrect result. So always add! Together we work through countless examples in detail, all while creating equivalent systems using the elimination method for solving systems of linear equations. Elimination Method (How-To) – VideoGet access to all the courses and over 450 HD videos with your subscription Monthly and Yearly Plans Available Get My Subscription Now The Elimination Method Learning Objective(s) · Solve a system of equations when no multiplication is necessary to eliminate a variable. · Solve a system of equations when multiplication is necessary to eliminate a variable. · Recognize systems that have no solution or an infinite number of solutions. · Solve application problems using the elimination method. Introduction The elimination method for solving systems of linear equations uses the addition property of equality. You can add the same value to each side of an equation. So if you have a system: x – 6 = −6 and x + y = 8, you can add x + y to the left side of the first equation and add 8 to the right side of the equation. And since x + y = 8, you are adding the same value to each side of the first equation. Using Addition to Eliminate a Variable If you add the two equations, x – y = −6 and x + y = 8 together, as noted above, watch what happens. You have eliminated the y term, and this equation can be solved using the methods for solving equations with one variable. Let’s see how this system is solved using the elimination method.
Unfortunately not all systems work out this easily. How about a system like 2x + y = 12 and −3x + y = 2. If you add these two equations together, no variables are eliminated. But you want to eliminate a variable. So let’s add the opposite of one of the equations to the other equation. 2x + y =12 → 2x + y = 12 → 2x + y = 12 −3x + y = 2 → − (−3x + y) = −(2) → 3x – y = −2 5x + 0y = 10 You have eliminated the y variable, and the problem can now be solved. See the example below.
The following are two more examples showing how to solve linear systems of equations using elimination.
Go ahead and check this last example—substitute (2, 3) into both equations. You get two true statements: 14 = 14 and 16 = 16! Notice that you could have used the opposite of the first equation rather than the second equation and gotten the same result. Using Multiplication and Addition to Eliminate a Variables Many times adding the equations or adding the opposite of one of the equations will not result in eliminating a variable. Look at the system below. 3x + 4y = 52 5x + y = 30 If you add the equations above, or add the opposite of one of the equations, you will get an equation that still has two variables. So let’s now use the multiplication property of equality first. You can multiply both sides of one of the equations by a number that will result in the coefficient of one of the variables being the opposite of the same variable in the other equation. This is where multiplication comes in handy. Notice that the first equation contains the term 4y, and the second equation contains the term y. If you multiply the second equation by −4, when you add both equations the y variables will add up to 0. 3x + 4y = 52 → 3x + 4y = 52 → 3x + 4y = 52 5x + y = 30 → −4(5x + y) = −4(30) → −20x – 4y = −120 −17x + 0y = −68 See the example below.
There are other ways to solve this system. Instead of multiplying one equation in order to eliminate a variable when the equations were added, you could have multiplied both equations by different numbers. Let’s remove the variable x this time. Multiply Equation A by 5 and Equation B by −3.
These equations were multiplied by 5 and −3 respectively, because that gave you terms that would add up to 0. Be sure to multiply all of the terms of the equation. Felix needs to find x and y in the following system. Equation A: 7y − 4x = 5 Equation B: 3y + 4x = 25 If he wants to use the elimination method to eliminate one of the variables, which is the most efficient way for him to do so? A) Add Equation A and Equation B B) Add 4x to both sides of Equation A C) Multiply Equation A by 5 D) Multiply Equation B by −1 Special Situations Just as with the substitution method, the elimination method will sometimes eliminate both variables, and you end up with either a true statement or a false statement. Recall that a false statement means that there is no solution. Let’s look at an example.
Graphing these lines shows that they are parallel lines and as such do not share any point in common, verifying that there is no solution.
If both variables are eliminated and you are left with a true statement, this indicates that there are an infinite number of ordered pairs that satisfy both of the equations. In fact, the equations are the same line.
Graphing these two equations will help to illustrate what is happening.
Solving Application Problems Using the Elimination Method The elimination method can be applied to solving systems of equations that model real situations. Two examples of using the elimination method in problem solving are shown below.
Summary Combining equations is a powerful tool for solving a system of equations. Adding or subtracting two equations in order to eliminate a common variable is called the elimination (or addition) method. Once one variable is eliminated, it becomes much easier to solve for the other one. Multiplication can be used to set up matching terms in equations before they are combined. When using the multiplication method, it is important to multiply all the terms on both sides of the equation—not just the one term you are trying to eliminate. Why does elimination work as a method of solving systems?The elimination method is one of the most widely used techniques for solving systems of equations. Why? Because it enables us to eliminate or get rid of one of the variables, so we can solve a more simplified equation.
How does elimination work in the systems of equations?The elimination method for solving systems of linear equations uses the addition property of equality. You can add the same value to each side of an equation. So if you have a system: x – 6 = −6 and x + y = 8, you can add x + y to the left side of the first equation and add 8 to the right side of the equation.
How does elimination method work?In the elimination method you either add or subtract the equations to get an equation in one variable. When the coefficients of one variable are opposites you add the equations to eliminate a variable and when the coefficients of one variable are equal you subtract the equations to eliminate a variable.
What are the advantages of using the elimination method?The advantages of using the elimination method are: The elimination method has fewer steps than other methods. It reduces the possibility of mistakes compared to other methods.
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Why does the elimination method work when solving a system of equations
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