Today

1
Todays Agenda

• E How do you rewrite equations from standard
  form to slope-intercept form?
• A - Warm-up Writing systems equations
• Journal Describe 2 ways to graph a line
• Review Looking at ymxb
• T - New Rewriting equations from standard to
  slope-intercept form (graphic organizer)
• Classwork Racing Game
• S Journal writing to answer LEQ
• Homework Rewriting Equations worksheet

2
JOURNAL TIME (8 mins) ?

• Directions Write down the journal prompt and
  then answer
• Describe the 2 different ways to graph a line
  that we have learned in class so far (Be
  specific)
• Which way do you prefer and why?

3
Rewriting Equations ?

• Lesson Essential Question
• How do you rewrite equations from standard form
  to graphing form?

4
2 Forms of Linear Equations ?

• The forms of linear equations are the formats in
  which the information is written in.
• These two forms are the most commonly used ways
  to write linear equations.
• 1. Standard Form Ax By C
• 2. Slope Intercept Form ymxb

5
Important!!! ?

• This is one of the BIG concepts you learned in
  Algebra I. You need to thoroughly understand
  this!
• Slope Intercept Form
• y mx b
• m represents the slope
• b represents the y-intercept

6
Review -Writing Equations Given Slope
Y-intercept

• Write the equation of a line that has a
  y-intercept of - 3 and a slope of - 4.
• y -3x 4
• y -4x 3
• y -3x 4
• y -4x 3

7
Review Find the slope and y-intercept of y 4
2x

• m 2 b 4
• m 4 b 2
• m -2 b 4
• m 4 b -2

8
Standard Form to Slope Intercept Form

• Ax By C
• to
• y mx b

9
What is Standard Form ?

• Standard Form is Ax By C
• Basically, if your x and y are on the same side
  of the equation, then it is in standard form.

10
Identify the equations in standard form

• A. 2x 4y 6
• B. y 3x 1
• C. x y 1
• D. 4y 5x 8
• E. -x 2y 6
• F. y 1/3x 2.5
• G. 2y 7 3x

11
Converting from standard form (Ax By C) to
slope-intercept form (y mx b)
12
Converting Standard to Slope-Intercept form

2x 3y 6
ax by c
(Standard Form)
-2x
-2x

3y 6 - 2x
y mx b
(Slope- Intercept)
13
3 Powerful Moves to get your equation into y

• 6X 4Y 12

1. MOVE X
2. DROP ALL
3. DIVIDE ALL

14
3 Powerful Moves to get your equation into y

• 6X 4Y 12

• 6X 4Y 12
• -6x -6x

• MOVE X
• Add or Subtract the x term to the other side of
  equals.

15
3 Powerful Moves to get your equation into y

• 6X 4Y 12

• 6X 4Y 12
• -6x -6x
• 4y 12 -6x

• 2. DROP ALL
• Bring all terms down in order.
• Do not add or subtract unlike terms!!!

16
3 Powerful Moves to get your equation into y

• 6X 4Y 12

• 6X 4Y 12
• -6x -6x
• 4y 12 -6x
• 4 4 4
• y 3 3/2 x

• 3. DIVIDE ALL
• Divide each term by the number attached to y
• keep slope a fraction!

17
3 Powerful Moves to get your equation into y

• -4X 3Y 12

1. MOVE X
2. DROP ALL
3. DIVIDE ALL

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3 Powerful Moves to get your equation into y

• -4X 3Y 12

• -4X 3Y 12
• 4x 4x

• MOVE X
• Add or Subtract the x term to the other side of
  equals.

19
3 Powerful Moves to get your equation into y

• -4X 3Y 12

• -4X 3Y 12
• 4x 4x
• 3y 12 4x

• 2. DROP ALL
• Bring all terms down in order.
• Do not add or subtract unlike terms!!!

20
3 Powerful Moves to get your equation into y

• -4X 3Y 12

• -4X 3Y 12
• 4x 4x
• 3y 12 4x
• 3 3 3
• y 4 4/3 x

• 3. DIVIDE ALL
• Divide each term by the number attached to y
• keep slope a fraction!

21
The 3 Power Moves to getting linesinto y form.

• 1.
• 2.
• 3.

MOVE the x term by Adding/Subtracting!
Drop ALL!
Divide ALL!
22
Pair Race Directions

• Equations are going to flash on the screen. The
  first one to step forward and describe the first
  step to converting the equation will earn the
  point.
• First person to answer correctly wins!
• Everyone.Please pay attention ?

23
Example

• 3x 2y 18

The first step is Subtract 3x from both sides
That would look like 3x 2y 18
-3x -3x
2y 18 3x
24
Example

• -7x 14y 28

The first step is Add 7x to both sides
That would look like -7x 14y 28
7x 7x
14y 28 7x
25
Lets Race!

• As quickly and quietly as possible line up
  please!
• No hitting, touching, pushing, pokingjust get in
  line!

Ready, Set, GO!!
26
4x 5y 10
Correct! Subtract 4x from both sides!! Great Job!
4x 5y 10
-4x -4x
5y 10 4x
27
-6x 3y 12
Correct! Add 6x to both sides!! Great Job!
-6x 3y 12
6x 6x
3y 12 6x
28
9x - y -8
Correct! Subtract 9x from both sides!! Great Job!
9x - y -8
-9x -9x
- y -8 9x
29
10x - 20y 20
Correct! Subtract 10x from both sides!! Great
Job!
10x - 20y 20
-10x -10x
-20y 20 10x
30
-11x 11y 33
Correct! Add 11x to both sides!! Great Job!
-11x 11y 33
11x 11x
11y 33 11x
31
-4x 2y 8
Correct! Add 4x to both sides!! Great Job!
-4x 2y 8
4x 4x
2y 8 4x
32
-8x - 4y -16
Correct! Add 8x to both sides!! Great Job!
-8x - 4y -16
8x 8x
-4y -16 8x
33
7x y -2
Correct! Subtract 7x from both sides!! Great Job!
7x y -2
-7x -7x
y -2 7x
34
2x 2y 10
Correct! Subtract 2x from both sides!! Great Job!
2x 2y 10
-2x -2x
2y 10 2x
35
-5x 3y -9
Correct! Add 5x to both sides!! Great Job!
-5x 3y -9
5x 5x
3y -9 5x
36
-8x - 4y 24
Correct! Add 8x to both sides!! Great Job!
-8x - 4y 24
8x 8x
-4y 24 8x
37
6x 12y -36
Correct! Subtract 6x from both sides!! Great Job!
6x 12y -36
-6x -6x
-12y -36 6x
38
-2x 2y -14
Correct! Add 2x to both sides!! Great Job!
-2x 2y -14
2x 2x
-2y -14 2x
39
GREAT GAME!!!! Please go back to your seats, we
are going to return to our notes and get this
first step written down and committed to memory ?
40
First StepExample Problem 1

• 6x 3y 9

41
First StepExample Problem 2

• -10x 2y 8

42
First StepExample Problem 3

• x - 2y 4

43
First StepExample Problem 4

• -x y -2

44
First StepExample Problem 5

• -8x 2y -2

45
Your Turn

• With your shoulder buddy, complete the 10
  problems on the next page.
• Remember, you are only showing the first step!
• You have 5 minutes to get this completed ?

46
Pair Race Directions

• Equations are going to flash on the screen. The
  first one to step forward and show the first AND
  second steps to converting the equation will win.
• Circle the slope and square on y-intercept
• What ever side of the room has the most points
  wins!
• EVERYONE.Please pay attention ?

47
Example

• 4x 2y 18

The first step is Subtract 4x from both sides
That would look like 4x 2y 18
-4x -4x
2y 18 4x
The second step is Divide everything by 2
2y 18 4x
2
2
2
Final Result y 9 2x
48
Example

• -14x 7y 28

The first step is Add 14x to both sides
That would look like -14x 7y 28
14x 14x
7y 28 14x
The second step is Divide everything by 7
7y 28 14x
7
7
7
Final Result y 4 2x
49
Example

• -8x 2y -10

The first step is Add 8x to both sides
That would look like -8x 2y -10
8x 8x
-2y -10 8x
The second step is Divide everything by -2
-2y -10 8x
-2
-2
-2
Final Result y 5 4x
50
Example

• 12x 6y 18

The first step is Subtract 12x from both sides
That would look like 12x 6y 18
-12x -12x
-6y 18 12x
The second step is Divide everything by -6
-6y 18 12x
-6
-6
-6
Final Result y -3 2x
51
Lets Race Again!
ReadySet GO!!
52
4x 5y 10
First Step? Correct! Subtract 4x from both
sides!!
4x 5y 10
-4x -4x
5y 10 4x
Second Step? Correct! Divide everything by 5!!
53
5y 10 4x
y 2 4/5x
54
-6x 3y 12
Correct! Add 6x to both sides!!
-6x 3y 12
6x 6x
3y 12 6x
Second Step? Correct! Divide everything by 3!!
55
3y 12 6x
y 4 2x
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9x - y -8
Correct! Subtract 9x from both sides!!
9x - y -8
-9x -9x
- y -8 9x
Whats in front of the ythat is always therewe
just dont write it (because mathematicians are
lazy ?)?
- 1y -8 9x
Second Step? Correct! Divide everything by -1!!
57
-y -8 9x
y 8 9x
58
10x - 20y 20
Correct! Subtract 10x from both sides!!
10x - 20y 20
-10x -10x
-20y 20 10x
Second Step? Correct! Divide everything by -20
59
-20y 20 10x
y -1 1/2x
60
-11x 11y 33
Correct! Add 11x to both sides!!
-11x 11y 33
11x 11x
11y 33 11x
Second Step? Correct! Divide everything by 11
61
11y 33 11x
y 3 x
62
-4x 2y 8
Correct! Add 4x to both sides!!
-4x 2y 8
4x 4x
2y 8 4x
Second Step? Correct! Divide everything by 2
63
2y 8 4x
y 4 2x
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-8x - 4y -16
Correct! Add 8x to both sides!!
-8x - 4y -16
8x 8x
-4y -16 8x
Second Step? Correct! Divide everything by -4!!
65
-4y -16 8x
y 4 - 2x
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7x y -2
Correct! Subtract 7x from both sides!!
7x y -2
-7x -7x
y -2 7x
Second Step? Correct! There is no second step!
Its already solved for y ?
67
y -2 - 7x
y -2 -7x
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2x 2y 10
Correct! Subtract 2x from both sides!!
2x 2y 10
-2x -2x
2y 10 2x
Second Step? Correct! Divide everything by 2!!
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2y 10 2x
y 5 -x
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-5x 3y -9
Correct! Add 5x to both sides!!
-5x 3y -9
5x 5x
3y -9 5x
Second Step? Correct! Divide everything by 3
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3y -9 5x
y -3 5/3x
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-8x - 4y 24
Correct! Add 8x to both sides!!
-8x - 4y 24
8x 8x
-4y 24 8x
Second Step? Correct! Divide everything by -4
73
-4y 24 8x
y -6 - 2x
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6x 12y -36
Correct! Subtract 6x from both sides!!
6x 12y -36
-6x -6x
-12y -36 6x
Second Step? Correct! Divide everything by -12
75
-12y -36 6x
y 3 1/2x
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-2x 2y -14
Correct! Add 2x to both sides!!
-2x 2y -14
2x 2x
-2y -14 2x
Second Step? Correct! Divide everything by -2
77
-2y -14 2x
y 7 - x
78
Putting it all TogetherFirst Second Step
Example Problem 1

• 35x 7y 49

79
Putting it all TogetherFirst Second Step
Example Problem 2

• -20x 5y -30

80
Putting it all TogetherFirst Second Step
Example Problem 3

• -6x 3y 24

81
Putting it all TogetherFirst Second Step
Example Problem 4

• -x 2y 4

82
Putting it all TogetherFirst Second Step
Example Problem 5

• x y 8

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Putting it all TogetherFirst Second Step
Example Problem 6

• x 4y 8

84
Your Turn

• With your shoulder buddy, complete the 10
  problems on the next page.
• Remember, you are completing the entire problem
  to solve for y.
• You have 10 minutes to get this completed ?

85
ERROR ANALYSIS
Four students rewrote the equation 12x 3y 9
into slope-intercept form. Determine who did it
correctly. If the student did it incorrectly,
explain the mistake.
Molly 12x 3y 9 3y 9 12x
y 3 12x
JARED 12x 3y 9 3y 9 12x
y 3 4x
Mia 12x 3y 9 3y 9 12x
y 3 4x y 4x - 3
Ali 12x 3y 9 4x y 3 y
-4x 3
86
Journal Time!!

• What are the three power moves that get any
  standard form equation into slope- intercept
  form?
• Write an example problem and rewrite it from
  standard form into slope-intercept form!

87
Homework!

• Complete the Slope Intercept and Standard Form
  wsht

88
Pick a partner activity (10 mins)

• Pick a partner within your color group to work on
  the problem
• Make sure that you work TOGETHER and CHECK EACH
  OTHERS WORK.
• This will be a graded assignment to earn bonus
  points on your quiz ?

89
You have 2 minutes to find your partner!
Purple Group Colette Dan Kayla Sydni Jonathan Phil
Pink Group Megan Tyheim Tiyana Alisa Courtney
Orange Group Daysia Taylor Chris
M Shiela Andy Chris N Ashley Steven
90
Purple Group

• Directions

• Finding X Y intercepts

• For the following problems find the x y
  intercepts. Show work!
• Dont forget that the x intercept happens when
  y0 and the y intercept happens when x0
• Write all intercepts as an ordered pair (x,y)

• a. 2x 3y 12
• b. 2x 3y 12
• c. 3x y 6
• d. y x 5

91
Orange Group

• Directions

• Rewriting Equations

• Rewrite each equation into slope-intercept form
  (y mxb)
• Identify the slope and y-intercept
• Show all work!

• 3x 2y 28
• b) 5y 15 2x
• c) 3y 9 2x

92
Pink Group

• Directions

• Rewriting Equations

• Rewrite each equation into slope-intercept form
  (y mxb)
• Identify the slope and y-intercept
• Dont forget the 3 POWER steps..Use your notes if
  needed!
• Show all work!

• a) x y 20
• 5x 4y 24
• 3x 2y 12

93
(No Transcript)
94
Solve Systems of Equations by the Graphing Method

• Lesson Essential Question
• Describe the types of solutions a system of
  equations can have?

95
What is a system of equations?

• A system of equations is when you have two or
  more equations using the same variables.
• The solution to the system is the point that
  satisfies ALL of the equations. This point will
  be an ordered pair.
• When graphing, you will encounter three
  possibilities.

96
Intersecting Lines (One Solution)

• The point where the lines intersect is your
  solution.
• What is the solution?
• The solution of this graph is (1, 2)

(1,2)
97
Find the solution to the following system using
the Graphing Method

• y -2x 4
• y x - 2
• Graph both equations. I will graph using
  slope-intercept form.
• Graph the y-intercept, then the slope.

y -2x 4 y int. (0, 4) and Slope -2/1 or
2/-1
y x - 2 y int. (0, -2) and Slope 1/1 or
-1/-1
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Step 2 Graph the equations.

• y -2x 4
• y x - 2
• Where do the lines intersect?
• (2, 0)

2x y 4
x y 2
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Step 3 Check your answer!

• To check your answer, plug the point back in for
  x and y into both equations and simplify.
• y -2x 4
• (0) -2(2) 4
• 0 -4 4
• 0 0
• y -x 2
• (0) -(2) 2
• 0 0

Nice joblets look how to solve it using the
graphing calculator!
100
Quick Stop Jot

• DO ALL LINES ALWAYS HAVE A POINT OF INTERSECTION?
• WHAT OTHER TYPES OF SOLUTIONS CAN SYSTEMS OF
  EQUATIONS HAVE?

101
Another type of solution

• How would you describe these lines?
• Y 3x 2
• Y 3x - 4
• What do you think the solution, or point of
  intersection, is?

102
Parallel Lines(No Solution)

• These lines never intersect!
• Since the lines never cross, there is NO
  SOLUTION!
• Parallel lines have the same slope with different
  y-intercepts.

103
Find the solution to the following system by the
Graphing Method

• y 2x 3
• y 2x 1
• Graph both equations using slope and
  y-intercept.

104
Step 2 Graph the equations.

• y 2x 3
• m 2 and b -3
• y 2x 1
• m 2 and b 1
• Where do the lines intersect?
• No solution!

Notice that the slopes are the same with
different y-intercepts. If you recognize
this early, you dont have to graph them!
105
Step 3 Check your answer!

• Not a lot to checkJust make sure you set up your
  equations correctly.
• I double-checked it and I did it right?

106
Another type of solution

• What do you notice about the graphs and
  equations?
• y -3x 4
• 3x y 4
• What do you think the solution, or point of
  intersection is?

107
Infinitely Many Solutions
SAME LINE
108
Coinciding Lines (Infinitely Many Solutions)

• These lines are the same!
• Since the lines are on top of each other, there
  are INFINITELY MANY SOLUTIONS!
• Coinciding lines have the same slope and
  y-intercepts.

109
Find the solution to the following system by the
Graphing Method

• Graph 6x 4y 12 and 3x 2y 6

110
JOURNAL Does it have a solution?
Determine whether the following have one, none,
or infinite solutions by looking at the slope
and y-intercept. Explain your reasoning.
1)
2)
3)
111
Does it have a solution?
Determine whether the following have one, none,
or infinite solutions by just looking at the
slope and y-intercepts.
1)
3)
2)
ANS One Solution
ANS No Solution
ANS Infinite Solutions
112
What is the solution of the system graphed below?

1. (2, -2)
2. (-2, 2)
3. No solution
4. Infinitely many solutions

113
What is the solution of this system using the
Graphing Method?
y 2x - 2 y 2x 1

1. (2, -2)
2. (2, 1)
3. No solution
4. Infinitely many solutions

114
What is the solution of this system using the
Graphing Method?
y 2x - 2 y 1/2x 4

1. (4, 6)
2. (6, 4)
3. No solution
4. Infinitely many solutions

115
What is the solution of this system using the
Graphing Method?
y 3x - 8 y 3x - 8

1. (3, 1)
2. (4, 4)
3. No solution
4. Infinitely many solutions

116
What is the solution of this system using the
Graphing Method?
y 4x - 2 -4x y -2

1. (4, -2)
2. (-2, 4)
3. No solution
4. Infinitely many solutions

117
Solving a system of equations by the Graphing
Method ?

• Let's summarize! There are 3 steps to solving a
  system using a graph.

Graph using slope and y intercept. Be sure to
use a ruler and graph paper!
Step 1 Graph both equations.
This is the solution! LABEL the solution (x, y)!
Step 2 Do the graphs intersect?
Substitute the x and y values into both equations
to verify the point is a solution to both
equations.
Step 3 Check your solution.
118
Summarize Time

• In your journals, write todays date and the
  question below.
• Describe systems of equations that have one
  solution, no solution, and infinitely many
  solutions?
• Include a graph and equations as examples.
• Answer the question in complete sentences with
  lots of details.

119
GRAPHING CALCULATOR

• Rewrite equation in y form
• Use the INTERSECT function to find the
  intersection point

120
(No Transcript)
121
Your turn GRAPHING EXAMPLES

• y - 3x and y 2 4x
• x y 1 and 2x y 4
• 3x y 1 and y 8 1/2x
• 2x y 1 and 5x 4y 10
• y 2x 3 and y -4 2x
• 6x 4y 12 and 3x 2y 6

122
GRAPHING CALCULATOR EXAMPLES
(2, - 6)

• y - 3x and y 2 4x
• x y 1 and 2x y 4
• 3x y 1 and y 8 1/2x
• 2x y 1 and 5x 4y 10
• y 2x 3 and y -4 2x
• 6x 4y 12 and 3x 2y 6

(3, -2)
(-2, 7)
(-2, 5)
No solution
Infinitely Many
123
All I Do Is Solve (Part I)
http//www.youtube.com/watch?vqxHCEwrpMw0NR1
124
Check Your Understanding

• Solve the system of equations using the Graphing
  Method. Check your solution.
• y 3x 3
• y -x 1

125
Group Self-Evaluation Form

• Read each statement and rate your partner by
  circling one response for each statement.

126
Homework Assignment

• Worksheet - Solve each system of equations by the
  Graphing Method.