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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.
Find the solution set for
(2, 7)
Intersecting lines because there is one solution.
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
x + y = 5 3x - y = 7 4x = 12 4 4
x = 3
𝐀𝐱+𝐁𝐲=𝐂
𝐀𝐝𝐝𝐢𝐭𝐢𝐨𝐧
y and -y
𝐱=𝟑
+
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=𝟐
Remember x = 3x + y = 5 3 + y = 5 -3 -3 y = 2
Find the solution set for
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.
(3, 2)
Intersecting lines because there is one solution.
Find the solution set for
Find the solution set for Procedures
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)
Both equations are already in Standard From
However, what is different about this system when in Standard Form?The terms do NOT have like coefficients.
4x + y = 3- x - 2y = 8
𝐀𝐱+𝐁𝐲=𝐂
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
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=−𝟓
Remember x = 24x + y = 3 4(2) + y = 3 8 + y = 3 -8 -8 y = -5
Find the solution set for
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?
(2, -5)
Intersecting lines because there is one solution.
Find the solution set for
ProceduresMake 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)
Both equations are already in Standard From
Notice none of the terms are equal again.
2x + 5y = -2210x + 3y = 22
𝐀𝐱+𝐁𝐲=𝐂
𝟏𝟎
x
-
5( )
10x + 25y = -11010 x + 3y = 22
y = -6
Decide what operation should now be used to eliminate the y term and complete
10x + 25y = -110 10x + 3y = 22 22y = -132 22 22
Find the solution set for
Procedures Substitute the value of the variable above into either original equation to solve for the remaining unknown variable. What did that variable equal?
x
Remember y = -62x + 5y = -22 2x + 5(-6) = -22 2x – 30 = -22 +30 +30 2x = 8 2 2 x = 4
Find the solution set for
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?
(4, -6)
Intersecting lines because there is one solution.
Find the solution set for
Find the solution set for Procedures
Make sure both equation 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)
Already in Standard
Form
𝟐
x x + y = 32x + 2y = 6
2( ) 2x + 2y = 62x + 2y = 6
Decide what operation should now be used to eliminate the y term and complete
2x + 2y = 6 2x + 2y = 6 0 = 0
y = -x + 3 +x + x x + y = 3
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?
There are infinite many solutions.
Coinciding lines because both sides of the equation are equal.
Find the solution set for
Procedures
Make sure both equation are in Standard Form:.
Pick a variable to eliminate:
What operation would cancel out 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)
Already in Standard Form
addition
x -3x + y = 1 3x – y = 6 0 = 7
Find the solution set for
+
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?
There are no solutions.
Parallel lines because 0 can never equal 7.
Find the solution set for