Elimination Method for Systems of Equations

Elimination

Systems of Equations - How to use the Elimination Method

What are we trying to do?

Find out where two equations intersect. This is used in business, government, etc.

How can we do this?

• Graphs - We can use graphs to see where lines intersect. To do this, you would have to use graph paper, and you would have to be able to easily graph lines. This can be cumbersome.

• Substitution - we can substitute one expression into another if either x or y equals the rest of an equation.

• Elimination - we can eliminate one variable, and then solve for the other to start finding where the lines intersect.

No matter what method we use, we have to know how to plug values into an equation.

If x + 2y = 11, and if y = 4, what would x be?

Find the value of x, if y = 4 in the following equation:

x + 2y = 11

• x + 2(4) = 11

• x + 8 = 11

• x = 11 - 8

• x = 3

Open Ended

Find the value of y, if x = 1 in the following equation.

3x + 2y = 11

How do you know which method to use?

If you have two equations that look like this, then you should use the substitution method.

y = 3x - 17

If you have two equations that look like this, then you should use the elimination method.

3x - 4y = 26

How do you eliminate?

1a. If the letters have the same number in front of them, but are opposite signs (+/-), then you just do what it says. For example:

3x + y = 12

5x - y = 4

8x = 16

x = 16/8

x = 2

How do you eliminate?

1b. If the letters have the same number in front of them, but are opposite signs (+/-), then you just do what it says. For example:

3x + 5y = 19

2x - 5y = -4

5x = 15

x = 15/5

x = 3

How do you eliminate?

1c. If the letters have the same number in front of them, and they have the same signs, then you have to change ALL signs in the second equation. For example:

4x + 3y = 2

2x +3y = 4......This would change to -2x - 3y = -4

4x + 3y = 2

-2x - 3y = -4

2x = -2

x = -2/2

x = -1

What if the numbers are NOT the same?

If there are different numbers in front of every letter, then you have to choose which one to eliminate.  Don't forget that when you know what letter, make sure you have different signs after you multiply.

2x + 3y = 10

5x + y = -1

Multiple Choice

Which variable (x or y) would you eliminate?

2x + 3y = 10

5x + y = -1

• x

• y

• neither

Multiple Choice

2x + 3y = 10

5x + y = -1

To eliminate the y variable, I would multiply the second equation by:

• -2

• -3

• -5

Multiple Choice

 $5x-2y=4$  

 $3x+y=9$  

To eliminate the variable, I would multiply the second equation by:

• 5

• -5

• 2

• -2

Multiple Choice

 $5x-2y=4$  

 $3x+y=9$  

After you multiply, the second equation would look like this:

• 3x + 2y = 9

• 6x + 2y = 18

• -3x - 2y = -9

• -6x - 2y = -18

Multiple Choice

Great! You just eliminated the y variable. Now you have to solve for x. What would x equal, if you had this left for the two equations:

Equation 1 is: 5x = 4

Equation 2 is: 6x = 18

Add the two equations and solve for x

• 2

• 22

• 14

• 3

Open Ended

What is the solution to this system of equations, based on what you now know. ***Remember, if you are having problems, ASK a question!!

 $5x+2y=4$  
 $2x-y=7$  
Just start by finding out which one to eliminate, then multiply the second equation by the right number. Solve for x, then solve for y.