Solving Systems of Equations by Elimination by Tammy Wallace Varina High School

Solving Systems of

Equations by Elimination

byTammy Wallace

Varina High School

Solving by Systems by Elimination

The Addition and Subtraction Properties of Equality can be used solve a system of equation. Using this method is called the ELIMINATION METHOD.

This is done by adding or subtracting the equation together to eliminate one variable.

From there, the remaining variable is solved for a specific value, which is then used to find the complete solution to the system.

Find the solution set for Procedures

Make sure both equation are in Standard Form:

(_____________)

While in standard form, which terms has the same coefficient? Including the operation in front of each term, what operation would cancel out those terms? Use this operation to eliminate the terms . Solve for the remaining variable. The remaining variable equal?

Both equations are already in Standard From

5x – 6y = -32 3x + 6y = 48 8x = 16 8 8

x = 2

𝐀𝐱+𝐁𝐲=𝐂

𝐀𝐝𝐝𝐢𝐭𝐢𝐨𝐧

−𝟔 𝐲 𝐚𝐧𝐝𝟔 𝐲

𝐱=𝟐

+

Procedures Substitute the value of the variable above into either original equation to solve for the remaining unknown variable. What did that variable equal?

𝐲=𝟕

Find the solution set for

Remember x = 23x + 6y = 48 3(2) + 6y = 48 6 + 6y = 48 -6 -6 6y = 42 6 6 y = 7

Procedures

a)What is/are the solutions to the system?

b)If graphed, what type of lines would this system form and how can you determine this WITHOUT graphing the system?

c)Graph the system with your calculator to verify the solution set is correct.


(2, 7)

Intersecting lines because there is one solution.


Make sure both equation are in Standard Form:

(_____________)

While in standard form, which terms has the same coefficient? Including the operation in front of each term, what operation would cancel out those terms? Use this operation to eliminate the terms . Solve for the remaining variable. The remaining variable equal?


x + y = 5 3x - y = 7 4x = 12 4 4

x = 3


𝐀𝐝𝐝𝐢𝐭𝐢𝐨𝐧

y and -y

𝐱=𝟑

+


𝐲=𝟐

Remember x = 3x + y = 5 3 + y = 5 -3 -3 y = 2


Procedures



c)Graph the system with your calculator to verify the solution set is correct.

(3, 2)




Make sure both equations are in Standard Form:

(_____________)

Pick a variable to eliminate:

What is the least common multiple of both coefficients of those terms?

Multiply each equation by a number so the chosen variable to eliminate can have the same coefficients as the LCD(you may only need to multiply

one equation)


However, what is different about this system when in Standard Form?The terms do NOT have like coefficients.

4x + y = 3- x - 2y = 8


𝟐

y

+

2( ) 8x + 2y = 6-x – 2y = 8

x = 2

Decide what operation should now be used to eliminate the y term and complete

8x + 2y = 6-x – 2y = 87x = 147 7


𝐲=−𝟓

Remember x = 24x + y = 3 4(2) + y = 3 8 + y = 3 -8 -8 y = -5


Procedures



(2, -5)



ProceduresMake sure both equations are in Standard Form:

(_____________)




one equation)


Notice none of the terms are equal again.

2x + 5y = -2210x + 3y = 22


𝟏𝟎

x

-

5( )

10x + 25y = -11010 x + 3y = 22

y = -6


10x + 25y = -110 10x + 3y = 22 22y = -132 22 22



x

Remember y = -62x + 5y = -22 2x + 5(-6) = -22 2x – 30 = -22 +30 +30 2x = 8 2 2 x = 4


Procedures



(4, -6)




Make sure both equation are in Standard Form:.




one equation)

Already in Standard

Form

𝟐

x x + y = 32x + 2y = 6

2( ) 2x + 2y = 62x + 2y = 6


2x + 2y = 6 2x + 2y = 6 0 = 0

y = -x + 3 +x + x x + y = 3

Procedures



There are infinite many solutions.

Coinciding lines because both sides of the equation are equal.


Procedures

Make sure both equation are in Standard Form:.


What operation would cancel out those terms?


one equation)

Already in Standard Form

addition

x -3x + y = 1 3x – y = 6 0 = 7


+

Procedures



There are no solutions.

Parallel lines because 0 can never equal 7.


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Solving Systems of Equations by Elimination by Tammy Wallace Varina High School