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How Do You Solve a System of Equations Using the Substitution Method?

Find the solution to the system of equations 2x–y=2 and 2x+y=6 using the substitution method.

Summary

  1. 'x' and 'y' are variables
  2. Solve the equations for 'y' so we can replace 'y' with something in terms of 'x' in one of them
  3. Replace the 'y' in y=-2x+6 with 2x-2
  4. We have one equation with just one variable, x, so we can solve for x
  5. 2 is the x-value for the solution
  6. Plug x=2 into 2x-y=2 to solve for 'y'
  7. 2 is the y-value for the solution
  8. Plug x=2 into 2x+y=6 to confirm that y=2 for both equations

Notes

    1. We want to solve our equations for 'y' so that we can replace 'y' with something else in one of them
    2. This will give us an equation with just one variable, 'x'
    1. Adding y to both sides gives us 2x=y+2
    2. Then subtracting 2 from both sides gives us y=2x-2
    1. Subtracting 2x from both sides gives us y=-2x+6
    1. Since we have both equations set equal to 'y', we can plug one equation into the other
    1. Since we have both equations set equal to 'y', we can plug one equation into the other
    1. Plug the first equation, y=2x-2, into the second equation wherever there's a 'y'
    2. The second equation is y=-2x+6
    3. We end up with 2x-2=-2x+6
    1. Adding 2x to both sides we'll end up with 4x-2=6
    2. Then we can add 2 to both sides to get 4x=8
    3. Dividing both sides by 4 gives us x=2
    1. Now that we know that x=2 in our solution, we can just plug 2 in for x into one of our original equations
    2. Then we can solve for y to get the y-value for our solution
    1. To find the y-value, we can plug x=2 into each of the original equations and solve for y
    2. Plug x=2 into 2x-y=2
    3. Multiply the 2's together, and you'll get 4-y=2
    4. Subtract 4 from both sides to get -y=-2
    5. Then multiply both sides by -1 to get y=2
    1. Remember, a solution to a system of equations must make BOTH equations true
    2. So we need to make sure it works in our second equation as well
    1. To find the y-value, we can plug x=2 into the second equation and solve for y
    2. Plug x=2 into 2x+y=6
    3. Multiply the 2's together, and you'll get 4+y=6
    4. Subtract 4 from both sides and you get y=2
    1. (2,2) is the only ordered pair you can plug into both 2x-y=2 and 2x+y=6 to get true statements for both