Download Solving Systems by Elimination

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
Elimination using Addition
Consider the system
x - 2y = 5
2x + 2y = 7
Lets add both equations
to each other
REMEMBER: We are trying to find the
Point of Intersection. (x, y)
Elimination using Addition
Consider the system
x - 2y = 5
+
2x + 2y = 7
Lets 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 = 7
= 12
3x
x=4 
Lets add both equations
to each other
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
4 - 2y = 5
- 2y = 1
y= 1 
2
Lets substitute x = 4 into this
equation.
Solve for y
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
4 - 2y = 5
- 2y = 1
y= 1 
2
Lets substitute x = 4 into this
equation.
Solve for y
1
ANS: (4, 2 )
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
= 21
x=3 
ANS: (3, y)
Elimination using Addition
Consider the system
3x + y = 14
Substitute x = 3 into this equation
4x - y = 7
3(3) + y = 14
9 + y = 14
y=5
ANS: (3, 5 )
NOTE: We use the Elimination Method, if we can immediately
cancel out two like terms.
Examples…
1.
2x + y = 5
3x  y = 15
ANS: (4, -3)
2.
6y + x = 11
2y  x = 5
ANS: (-1, 2)
Elimination using
Multiplication
Consider the system
6x + 11y = -5
6x + 9y = -3
Elimination using
Multiplication
Consider the system
+
6x + 11y = -5
6x + 9y = -3
12x + 20y = -8
When we add equations together,
nothing cancels out
Elimination using
Multiplication
Consider the system
6x + 11y = -5
6x + 9y = -3
Elimination using
Multiplication
Consider the system
-1 (6x + 11y = -5 )
6x + 9y = -3
Elimination using
Multiplication
Consider the system
+
- 6x - 11y = 5
6x + 9y = -3
-2y = 2
y = -1 
ANS: (x, -1 )
Elimination using
Multiplication
Consider the system
6x + 11y = -5
6x + 9y = -3
Lets substitute y = -1 into this equation
y = -1
6x + 9(-1) = -3
6x + -9 = -3
+9 +9
6x = 6
x=1
ANS: (x, -1 )
Elimination using
Multiplication
Consider the system
6x + 11y = -5
6x + 9y = -3
Lets substitute y = -1 into this equation
y = -1
6x + 9(-1) = -3
6x + -9 = -3
+9 +9
6x = 6
x=1
ANS: ( 1, -1 )
Elimination using
Multiplication
Consider the system
x + 2y = 6
3x + 3y = -6
Multiply by -3 to eliminate the x term
Elimination using
Multiplication
Consider the system
-3 ( x + 2y = 6 )
3x + 3y = -6
Elimination using
Multiplication
Consider the system
+
-3x + -6y = -18
3x + 3y = -6
-3y = -24
y=8 
ANS: (x, 8)
Elimination using
Multiplication
Consider the system
x + 2y = 6
3x + 3y = -6
Substitute y =14 into equation
y =8
x + 2(8) = 6
x + 16 = 6
x = -10 
ANS: (x, 8)
Elimination using
Multiplication
Consider the system
x + 2y = 6
3x + 3y = -6
Substitute y =14 into equation
y =8
x + 2(8) = 6
x + 16 = 6
x = -10 
ANS: (-10 , 8)
Examples
1.
x + 2y = 5
2.
x + 2y = 4
2x + 6y = 12
x - 4y = 16
ANS: (3, 1)
ANS: (8, -2)
More complex Problems
Consider the system
3x + 4y = -25
2x - 3y = 6
Multiply by 2
Multiply by -3
More complex Problems
Consider the system
2( 3x + 4y = -25 )
-3( 2x - 3y = 6)
More complex Problems
Consider the system
+
6x + 8y = -50
-6x + 9y = -18
17y = -68
y = -4 
ANS: (x, -4)
More complex Problems
Consider the system
3x + 4y = -25
2x - 3y = 6
2x - 3(-4) = 6
2x - -12 = 6
2x + 12 = 6
2x = -6
Substitute y = -4
x = -3 
ANS: (x, -4)
More complex Problems
Consider the system
3x + 4y = -25
2x - 3y = 6
2x - 3(-4) = 6
2x - -12 = 6
2x + 12 = 6
2x = -6
Substitute y = -4
x = -3 
ANS: ( -3 , -4)
Examples…
2.
1.
4x + y = 9
2x + 3y = 1
3x + 2y = 8
5x + 7y = 3
ANS: (2, 1)
ANS: (2, -1)