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Let’s Warm Up!1) Solve the system of equations by graphing:
2x + 3y = 12 2x – y = 4
Answer:
2) Find the slope-intercept form for the equation of a line that passes through (0, 5) and is parallel to a line whose equation is 4x – y = 3?
Answer:
3)Solve 3│x – 5│= 12Answer:
(3, 2)
y=4x+5
x= 1, 9
Let’s chat about finals
Wednesday Jan 23rd: 2, 4, 6 Thursday Jan 24th: 1, 3, 5
Minimum Days
Final Review sheet due DAY OF FINAL
Extra Credit: “Additional Practice Problems”
Mini Quiz Time!
3 graphing Questions Get out a pencil please.
8-2 Substitution
Objective: To use the substitution method to solve systems of equations.
Two Algebraic Methods:
Substitution Method Elimination Method will learn
about next
RECALL…Three Types of Solutions: Intersection is Solution
One Solution No Solution
Infinite SolutionsSame slope
Different y-intercept“Run parallel Never
intersect”
Same slope Same y-intercept
“Same line Intersect infinitely”
Different slopeDifferent y-
intercept“Intersect at one
point”
Substitution Method
Use the substitution method when: one equation is set equal to a variable
y = 2x + 1 or x = 3y - 2
Example 1
Instead of x = 2 we have:x = y + 2x + 2y = 11
(y + 2) + 2y = 113y + 2 = 113y = 9y = 3
These are all the same!
x = 3 + 2x = 5
Answer: (5,3)
Try with a Mathlete
1) y = 3x x + 2y = -21
2) y = 2x – 6 3x + 2y = 9
Answers:1) (-3,-9)2) (3,0)
Example 2
x + 4y = 12x – 3y = -9
First, solve for a variablex = -4y + 1
2(-4y + 1) – 3y = -9-8y + 2 – 3y = -9-11y + 2 = -9-11y = -11y = 1
x = -4(1) + 1x = -3
Answer: (-3,1)
Solve for x (because there is no number in front of it)
TOO
1) 2y = -3x 2) 2x – y = -4 4x + y = 5 -3x + y = -
9
Answers:1) (2,-3)2) (13,30)
Special Cases
x + y = 16 x = 16 – y2y = -2x + 2
2y = -2(16 – y) + 22y = -32 + 2y + 22y = -30 + 2y0 = -30FalseNO SOLUTION
6x – 2y = -4 y = 3x + 2
6x – 2(3x + 2) = -46x – 6x – 4 = -4-4 = -4TrueINFINITELY MANY
TOO for Homework
1) y = -x + 3, 2y + 2x = 4
2) x + y = 0, 3x + y = -8
3) y = 3x – 7, 3x – y = 7
Homework
Pg. 467 #17-32 left column
MORE Explanations
The following slides have more examples and explanations of the substitution method.
Examples: Use the substitution method to solve the system of equations.
1) 2x + 3y = 2 x – 3y = –17
x – 3y = –17 +3y +3y x = 3y – 17
2x + 3y = 2 “x = 3y – 17” 2(3y – 17) + 3y = 2 6y – 34 + 3y = 2 –34 + 9y = 2 +34 +34 9y = 36
9 9 y = 4
x = 3y – 17 “y = 4” x = 3(4) – 17 x = 12 – 17 x = –5
1st: Transform one equation to isolate a variable
2nd: Substitute into the other equation and solve for variable #1
3rd: Substitute into transformed equation from 1st step and solve for variable #2
(use the substitution method when a variable is already isolated or when a variable has a coefficient of 1 and can easily be transformed)
Write answer as an ordered pair (x, y): One Solution(–5 , 4)
(we picked x – 3y = – 17 because x has a coefficient of 1 and can easily be transformed)
Examples: Use the substitution method to solve the system of equations.
2) –9x + 3y = –21 3x – y = 7
3x – y = 7-3x -3x –y = –3x + 7 -1 -1 -1 y = 3x – 7
–9x + 3y = –21 “y = 3x – 7” –9x + 3(3x – 7) = –21 –9x + 9x – 21= –21 –21 = –21
1st: Transform one equation to isolate a variable
2nd: Substitute it into the other equation and solve for variable #1
3rd: Substitute into the transformed equation from 1st step and solve for variable #2
(use the substitution method when a variable is already isolated or when a variable has a coefficient of 1 and can easily be transformed)
Write answer as an ordered pair (x, y): Infinite Solutions
(we picked 3x – y = 7 because y has a coefficient of -1 and can easily be transformed)
True!!
Examples: Use the substitution method to solve the system of equations.
3) 4x – 2y = 5 y = 2x + 1
4x – 2y = 5 “y = 2x + 1” 4x – 2(2x + 1) = 5 4x – 4x – 2 = 5 – 2 = 5
1st: Transform one equation to isolate a variable
2nd: Substitute into the other equation and solve for variable #1
3rd: Substitute into transformed equation from 1st step and solve for variable #2
(use the substitution method when a variable is already isolated or when a variable has a coefficient of 1 and can easily be transformed)
No Solution
y = 2x + 1
(already isolated)
False!!