Let

1
Lets 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 3x 5 12
• Answer

(3, 2)
y4x5
x 1, 9
2
Lets chat about finals

• Wednesday Jan 22nd 2, 4, 6
• Thursday Jan 23rd 1, 3, 5
• Minimum Days
• Final Review sheet
• due DAY OF FINAL

3
Mini Quiz Time!

• 3 graphing Questions
• Get out a pencil please.

4
8-2 Substitution

• Objective To use the substitution method to
  solve systems of equations.

5
Two Algebraic Methods

• Substitution Method
• Elimination Method ? will learn about next ?

6
RECALLThree Types of Solutions
? Intersection is Solution
Infinite Solutions
One Solution
No Solution
Same slope Same y-intercept Same line ?
Intersect infinitely
Same slope Different y-intercept Run parallel ?
Never intersect
Different slope Different y-intercept Intersect
at one point
7
Substitution Method

• Use the substitution method when
• one equation is set equal to a variable
• y 2x 1 or x 3y - 2

8
Example 1

• Instead of x 2 we have
• x y 2
• x 2y 11
• (y 2) 2y 11
• 3y 2 11
• 3y 9
• y 3

These are all the same!
x 3 2 x 5
Answer (5,3)
9
Try with a Mathlete

• y 3x
• x 2y -21
• y 2x 6
• 3x 2y 9
• Answers
• 1) (-3,-9)
• 2) (3,0)

10
Example 2

• x 4y 1
• 2x 3y -9
• First, solve for a variable
• x -4y 1
• 2(-4y 1) 3y -9
• -8y 2 3y -9
• -11y 2 -9
• -11y -11
• y 1

Solve for x (because there is no number in front
of it)
x -4(1) 1 x -3
Answer (-3,1)
11
TOO

• 2y -3x 2) 2x y -4
• 4x y 5 -3x y -9
• Answers
• 1) (2,-3)
• 2) (13,30)

12
Special Cases

• x y 16 x 16 y
• 2y -2x 2
• 2y -2(16 y) 2
• 2y -32 2y 2
• 2y -30 2y
• 0 -30
• False
• NO SOLUTION

• 6x 2y -4
• y 3x 2
• 6x 2(3x 2) -4
• 6x 6x 4 -4
• -4 -4
• True
• INFINITELY MANY

13
TOO for Homework

• 1) y -x 3, 2y 2x 4
• 2) x y 0, 3x y -8
• 3) y 3x 7, 3x y 7

14
Homework

• Pg. 467 17-32 left column

15
Pg. 467 17-32 Left ColumnSolve using
substitution.
17.
26.
20.
29.
23.
32.
16
MORE Explanations

• The following slides have more examples and
  explanations of the substitution method.

17
Examples Use the substitution method to solve
the system of equations.
1) 2x 3y 2 x 3y 17
(use the substitution method when a variable is
already isolated or when a variable has a
coefficient of 1 and can easily be transformed)
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
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
x 3y 17 3y 3y x 3y 17
(we picked x 3y 17 because x has a
coefficient of 1 and can easily be transformed)
One Solution (5 , 4)
Write answer as an ordered pair (x, y)
18
Examples Use the substitution method to solve
the system of equations.
2) 9x 3y 21 3x y 7
(use the substitution method when a variable is
already isolated or when a variable has a
coefficient of 1 and can easily be transformed)
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
9x 3y 21 y 3x 7 9x
3(3x 7) 21 9x 9x 21 21
21 21
3x y 7 -3x -3x y 3x 7
-1 -1 -1 y 3x 7
True!!
(we picked 3x y 7 because y has a
coefficient of -1 and can easily be transformed)
Infinite Solutions
Write answer as an ordered pair (x, y)
19
Examples Use the substitution method to solve
the system of equations.
3) 4x 2y 5 y 2x 1
(use the substitution method when a variable is
already isolated or when a variable has a
coefficient of 1 and can easily be transformed)
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
4x 2y 5 y 2x 1
4x 2(2x 1) 5 4x 4x 2 5
2 5
y 2x 1
(already isolated)
False!!
No Solution
20
Chapter 8 Systems of Equations 8-2 Substitution

• We will become experts at solving systems of
  equations with substitution.

21
Math Lab

• Solve the system with substitution
• x y 2
• x 2y 11
• (y 2) 2y 11
• 3y 2 11
• 3y 9
• y 3

These are all the same!
x 3 2 x 5
Answer (5,3)
22
Math Lab ReviewSubstitution

• y -x 3
• 2y 2x 4

• y 3x 7
• 3x y 7

• x y 0
• 3x y -8

23
Special Cases

• x y 16 x 16 y
• 2y -2x 2
• 2y -2(16 y) 2
• 2y -32 2y 2
• 2y -30 2y
• 0 -30
• False
• NO SOLUTION

• 6x 2y -4
• y 3x 2
• 6x 2(3x 2) -4
• 6x 6x 4 -4
• -4 -4
• True
• INFINITELY MANY

24
Partner Big Yellow Dice Game