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How Do You Solve a System of Equations Using the Elimination by Subtraction Method?

Find the solution to the following system of equations: 2x-y=2 and 2x+y=6

Summary

  1. You have to distribute the negative sign into the second equation before combining
  2. You can align each term in each equation to help you combine the two equations
  3. You can plug in the y-value we just got to find the correct x-value
  4. The first equation yields x=2 when we plugged 2 in for y
  5. The second equation also gave us x=2 when we plugged 2 in for y, which means the solution to the system is (2,2)

Notes

    1. You generally won't use both addition and subtraction together, but you can use either operation along side multiplication
    2. In this tutorial we'll just be using subtraction, and nothing else.
    1. You could use addition and multiplication, or just subtraction by itself!
    1. The system of equations are the equations 2x-y=2 and 2x+y=6
    1. The system of equations are the equations 2x-y=2 and 2x+y=6
    1. This is what the elimination method is all about!
    2. We subtract or add a multiple of one equation to the other equation to cancel out one of the variables
    1. This is what the elimination method is all about!
    2. We subtract or add a multiple of one equation to the other equation to cancel out one of the variables
    1. This is called a system of equations because there are two equations with two variables.
    1. Our equations are 2x-y=2 and 2x+y=6
    2. This is called a system of equations because there are two equations with two variables.
    1. You could solve this by addition, but it requires trying to cancel out the y instead of the x
    1. In other words, 2x + 2x = 4x
    2. Which means the x variable does not cancel out
    3. But if we had 2x - 2x instead, they WOULD cancel
    1. We're going to subtract 2x+y=6 from 2x-y=2
    1. So -(2x+y=6) should become -2x-y=-6
    1. We can take the positive 2x and align it above the -2x
    2. Align the two -y's above one another
    3. Finally, align the 2 above the -6 on the right hand side
    1. We're combining 2x-y=2 and -2x-y=-6
    2. When we combine term by term, we get:
    3. 2x-2x=0
    4. -y-y=-2y
    5. 2-6=4
    1. We've eliminated the x variable!
    1. Solving for y:
    2. -2y/(-2) = y
    3. -4/(-2) = 2
    1. We can plug in the y-value we just found to find the correct x-value
    1. The value we found for y is 2
    2. Our first equation is 2x-y-2
    3. Our second equation is 2x+y=6
    1. Our first equation is 2x-y-2
    1. We're trying to find the correct x-value
    1. Our first equation is 2x-y-2
    2. We plugged in the y-value we found and then solved for x
    1. In other words, if x=2 is really part of the solution, then it should satisfy both equations
    1. Our second equation is 2x+y=6
    1. Our second equation is 2x+y=6
    2. We plugged in the y-value we found and then solved for x
    1. This is what we were hoping to find, that both equations gave us the same x-value!
    1. Our solution has an x-value of 2 and a y-value of 2
    2. Those values give us an ordered pair of (2,2)
    1. Namely, we'd get 2=2 from the first equation, and 6=6 from the second equation