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Solving Systems of Equations The Elimination Method
Solving Systems of Equations
The Elimination Method
Objectives• Learn the procedure of the
Elimination Method using addition• Learn the procedure of the
Elimination Method using multiplication
• Solving systems of equations using the Elimination Method
SOLVING BY ELIMINATION***Line up like terms vertically between the two
equations before starting elimination**
•Step 1: Choose a variable to eliminate•Step 2: Eliminate that variable by adding, subtracting one equation from the other. (Sometimes you have to multiply first)•Step 3: Solve the new equation•Step 4: Plug in your answer to find the other variable•Step 5: Check your answer
Elimination using Addition
Consider the system
x - 2y = 5
2x + 2y = 7
REMEMBER: We are trying to find the Point of Intersection. (x, y)
Lets add both equations to each other
Elimination using Addition
Consider the system
x - 2y = 5
2x + 2y = 7Lets add both equations to each other+
NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.
Elimination using Addition
Consider the system
x - 2y = 5
2x + 2y = 7Lets add both equations to each other+
3x = 12x = 4
ANS: (4, y)
NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.
Elimination using Addition
Consider the system
x - 2y = 5
2x + 2y = 7
ANS: (4, y)
Lets substitute x = 4 into this equation.
4 - 2y = 5 Solve for y - 2y = 1
y = 12
NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.
Elimination using Addition
Consider the system
x - 2y = 5
2x + 2y = 7
ANS: (4, )
Lets substitute x = 4 into this equation.
4 - 2y = 5 Solve for y - 2y = 1
y = 12
12
NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.
Elimination using Addition
Consider the system
3x + y = 14
4x - y = 7
NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.
Elimination using Addition
Consider the system
3x + y = 14
4x - y = 7
7x = 21x = 3
ANS: (3, y)
+
Elimination using Addition
Consider the system
ANS: (3, )
3x + y = 14
4x - y = 7
Substitute x = 3 into this equation
3(3) + y = 149 + y = 14
y = 5
5
NOTE: We use the Elimination Method, if we can immediately cancel out two like terms.
Your Turn…
2x y+ 5=3x y 15=
1. 2.
2y x 5=6y x+ 11=
ANS: (4, -3) ANS: (-1, 2)
One more example4x = -8y + 20-4x + 2y = -30
Elimination using Multiplication
Consider the system
6x + 11y = -56x + 9y = -3
Elimination using Multiplication
Consider the system
6x + 11y = -56x + 9y = -3+
12x + 20y = -8 When we add equations together, nothing cancels out
Elimination using Multiplication
Consider the system
6x + 11y = -56x + 9y = -3
Elimination using Multiplication
Consider the system
6x + 11y = -56x + 9y = -3
-1 ( )
Elimination using Multiplication
Consider the system
- 6x - 11y = 56x + 9y = -3+
-2y = 2y = -1
ANS: (x, )-1
Elimination using Multiplication
Consider the system
6x + 11y = -5
6x + 9y = -3
ANS: (x, )-1
y = -1
Lets substitute y = -1 into this equation
6x + 9(-1) = -36x + -9 = -3
+9 +9
6x = 6x = 1
Elimination using Multiplication
Consider the system
6x + 11y = -5
6x + 9y = -3
ANS: ( , )-1
y = -1
Lets substitute y = -1 into this equation
6x + 9(-1) = -36x + -9 = -3
+9 +9
6x = 6x = 1
1
Elimination using Multiplication
Consider the system
x + 2y = 63x + 3y = -6
Multiply by -3 to eliminate the x term
Elimination using Multiplication
Consider the system
x + 2y = 63x + 3y = -6
-3 ( )
Elimination using Multiplication
Consider the system
-3x + -6y = -183x + 3y = -6+
-3y = -24y = 8
ANS: (x, 8)
Elimination using Multiplication
Consider the system
x + 2y = 63x + 3y = -6
ANS: (x, 8)
Substitute y =8 into equation
y =8x + 2(8) = 6
x + 16 = 6x = -10
Elimination using Multiplication
Consider the system
x + 2y = 63x + 3y = -6
ANS: ( , 8)
Substitute y =8 into equation
y =8x + 2(8) = 6
x + 16 = 6x = -10
-10
Examples1.
x + 2y = 5
2x + 6y = 12
2.
ANS: (3, 1)
x + 2y = 4
x - 4y = 16
ANS: (8, -2)
More complex ProblemsConsider the system
3x + 4y = -252x - 3y = 6
Multiply by 2Multiply by -3
More complex ProblemsConsider the system
3x + 4y = -25
2x - 3y = 6
2( )
-3( )
More complex ProblemsConsider the system
6x + 8y = -50-6x + 9y = -18+
17y = -68y = -4
ANS: (x, -4)
More complex ProblemsConsider the system
3x + 4y = -252x - 3y = 6
ANS: (x, -4)
Substitute y = -42x - 3(-4) = 6
2x - -12 = 62x + 12 = 6
2x = -6x = -3
More complex ProblemsConsider the system
3x + 4y = -252x - 3y = 6
ANS: ( , -4)
Substitute y = -42x - 3(-4) = 6
2x - -12 = 62x + 12 = 6
2x = -6x = -3
-3
Examples…1. 2.
4x + y = 9
3x + 2y = 8
2x + 3y = 1
5x + 7y = 3
ANS: (2, 1) ANS: (2, -1)
