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Systems of Equations (Elimination)

Here you’ll learn the final method for solving a system of equations. Below you can see instructions on how to proceed, followed by an example of how to solve a system of equations with this method.

Rule

The Elimination Method

1.
Choose one of the variables that you want to eliminate.
2.
Multiply the equations I and II with the numbers that make the variable you chose have the same coefficient, but with opposite signs.
3.
Write the new equations below the first two.
4.
Add equation II to I and write the answer below these equations. Now you’ve got one equation with one unknown. Solve it.
5.
Put the answer you find into equation I and solve for the final variable.
6.
Write your answer with coordinates: ANSWER: (x,y) = (a,b)

Example 1

Solve the system of equations I y + 2x = 1 II 2y x = 2

1.
I decide to get rid of y.
2.
Since you have 2y in II and y in I, you have to multiply I with 2 to eliminate y. In this case you don’t have to multiply equation II with anything: Iy + 2x = 1 |× ( 2) II2y x = 2̲
3.
Write the two equations again after applying the changes: 2y 4x = 2 2y 4 x̲ = 2̲
4.
Add the two equations together such that y is eliminated, and solve for x: 2y + 2y 4x x = 2 + 2 0y 5x = 0 x = 0
5.
Put the answer into either equation I or II. Here you can choose whichever you like. I choose equation I: y + 2x = 1 y + 2 × (0) = 1 y + 0 = 1 y = 1
6.
Write the answer with coordinates: ANSWER: (x,y) = (0, 1)

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