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Elimination method review (systems of linear equations)

The elimination method is a technique for solving systems of linear equations. This article reviews the technique with examples and even gives you a chance to try the method yourself.

What is the elimination method?

The elimination method is a technique for solving systems of linear equations. Let's walk through a couple of examples.

Example 1

We're asked to solve this system of equations:
2y+7x=55y7x=12
We notice that the first equation has a 7x term and the second equation has a 7x term. These terms will cancel if we add the equations together—that is, we'll eliminate the x terms:
2y+7x=5+ 5y7x=127y+0=7
Solving for y, we get:
7y+0=77y=7y=1
Plugging this value back into our first equation, we solve for the other variable:
2y+7x=521+7x=52+7x=57x=7x=1
The solution to the system is x=1, y=1.
We can check our solution by plugging these values back into the original equations. Let's try the second equation:
5y7x=12517(1)=?125+7=12
Yes, the solution checks out.
If you feel uncertain why this process works, check out this intro video for an in-depth walkthrough.

Example 2

We're asked to solve this system of equations:
9y+4x20=07y+16x80=0
We can multiply the first equation by 4 to get an equivalent equation that has a 16x term. Our new (but equivalent!) system of equations looks like this:
36y16x+80=07y+16x80=0
Adding the equations to eliminate the x terms, we get:
36y16x+80=0+ 7y+16x80=029y+00=0
Solving for y, we get:
29y+00=029y=0y=0
Plugging this value back into our first equation, we solve for the other variable:
36y16x+80=036016x+80=016x+80=016x=80x=5
The solution to the system is x=5, y=0.
Want to see another example of solving a complicated problem with the elimination method? Check out this video.

Practice

Problem 1
Solve the following system of equations.
3x+8y=152x8y=10
x=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
y=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Want more practice? Check out these exercises:

Want to join the conversation?

  • leaf green style avatar for user Olivia
    what if you are using three variables and you have three different equations?
    for example: (three variables,three equations)
    x-y-2z=4
    -x+2y+z=1
    -x+y-3z=11
    (55 votes)
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    • cacteye green style avatar for user Hamza Usman
      First of all, the only way to solve a question with 3 variables is with 3 equations. Having 3 variables and only 2 equations wouldn't allow you to solve for it. To start, choose any two of the equations. Using elimination, cancel out a variable. Using the top 2 equations, add them together. That results in y-z=5. Now, look at the third equation and cancel out the same variable that you originally cancelled out. In this case, we canceled out x. Adding the first equation to the 3rd equation would get rid of his. Adding would give -5z=15. We got lucky because both the x and the y cancelled out. If they didn't both cancel out, you would just have t solve the two equations which you should know how to do. Back to the problem, -5z=15, so z=-3.Plug that into the equation y-z=5 to solve for y. y-(-3)=5, so y+3=5. That gives y=2. Plug both of those into any of the three original equations and solve for x. You get x=0. Your final solution is x=0, y=2, and z=-3 or (0,2,-3).
      (49 votes)
  • aqualine seedling style avatar for user 8008161
    What if the numbers before x and y can not make up?

    4x-3y=8
    5x-2y=-11
    (6 votes)
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    • duskpin ultimate style avatar for user Dominic Nguyen
      I don't completely understand what you mean, but I have an idea of what you're asking. You can multiply both equations by a number to get one of the x or y absolute values the same, multiply the top equation by 2, to get 8x-6y=16, and the second equation by -3 to get -15x+6y=33, then add the equations to get -7x=49, so x is equal to -7. Plug in -7 for x to solve for y which would be -12
      Hope this helps
      (21 votes)
  • blobby green style avatar for user Yeyka Rosario
    what about unsorted equations?
    -4y-11x=36
    20=-10x-10y
    (7 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user charlietsmith1010
    what if you have -14x + 9y = 46 and 14x - 9y = 102. Elimination will cancel both x and y out. What do you do?
    (2 votes)
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    • stelly blue style avatar for user Kim Seidel
      You are correct, both the X and Y cancel out leaving you with: 0=148. This is a false statement. It is telling you that the system has no solution. It also means that the 2 lines are parallel. Parallel lines have no points in common which is why the system has no solution.
      Hope this helps.
      (13 votes)
  • blobby green style avatar for user kesvibp0208
    kesvi patel
    are we supposed to do last divide step ?
    (4 votes)
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    • scuttlebug yellow style avatar for user Rin
      After you add/subtract the new equations, you eliminate one of the variables and divide. After solving one of them, plug your solved variable to one of the original problems.
      This might help you understand more clearly:

      12x + 2y = 90 ... (1)
      6x + 4y = 90 ... (2)

      (2)*2 12x + 8y = 180 ... (2)'

      (1)-(2)' 12x + 2y = 90
      - 12x + 8y = 180
      -6y = -90
      You eliminated x
      and now you solve y by dividing.
      y = 15
      y = 15 plugged into (2). (Always pick the easier problem to solve your problem accurately.)
      6x + 4(15) = 90
      6x + 60 = 90
      6x = 90 - 60
      6x = 30
      x = 5

      Your final answer will be
      (x,y) = (5,15)

      Hope this helps!
      (8 votes)
  • duskpin ultimate style avatar for user Noa B
    I don't understand how we can do anything to the equasions 10y - 11x = -4 and -2y + 3x = 4!!
    and every time I do this, it marks me wrong for trying to multiply the 3 and -11. And the hint doesn't help either!
    (2 votes)
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    • stelly blue style avatar for user Kim Seidel
      You should multiply by 3 and +11 to create -33x and +33x. You need opposite signs to eliminate the x. By using -11, you are making the x terms have the same sign and when you add the 2 equations, no variable is eliminated.

      Try that and see how it works out.
      Comment back with your work if you still have issues.
      (9 votes)
  • blobby green style avatar for user Zainab
    Do you always have to eliminate the x term?
    (3 votes)
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  • female robot ada style avatar for user alise.blanton
    In an honors class in 8th grade, I am generally confused. Can someone maybe give me the steps in a simple worded form?
    (3 votes)
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    • leafers ultimate style avatar for user Evangeline Viklund
      Well, it depends on what you need. lets go with the equations:

      10y-11x=-4
      -2y+3x=4

      For this, you want to first make the bottom equation have an equivalent value for one of the variables so that we can eliminate it. the bottom equation is perfect, because:

      -2y(5)+3x(5)=4(5)
      this makes -10y+15x=20

      which in that case...

      10y-11x=-4
      -10y+15x=20
      =
      4x=16
      4x/4=16/4
      x=4

      from there, substitute 4 for x to solve for y, and you have your coordinate variables.
      (5 votes)
  • purple pi purple style avatar for user Omelette Kai
    when plugging in... do i plug in to the original equation or the (new) equivalent equation?
    (3 votes)
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  • scuttlebug yellow style avatar for user pineapple282
    Why can't you subtract one equation from another to eliminate variables?
    (2 votes)
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    • stelly blue style avatar for user Kim Seidel
      You can. You would get the same result if it's done correctly. A common error when subtracting the equations is that some signs get changed and others don't. So, the wrong result is created. Most people get fewer errors if they set up the signs first and then add the equations.
      (4 votes)