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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)