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Title: 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

18
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
56
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
64
-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
66
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
68
2x 2y 10
Correct! Subtract 2x from both sides!!
2x 2y 10
-2x -2x
2y 10 2x
Second Step? Correct! Divide everything by 2!!
69
2y 10 2x
y 5 -x
70
-5x 3y -9
Correct! Add 5x to both sides!!
-5x 3y -9
5x 5x
3y -9 5x
Second Step? Correct! Divide everything by 3
71
3y -9 5x
y -3 5/3x
72
-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
74
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
76
-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

83
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
98
Step 2 Graph the equations.
  • y -2x 4
  • y x - 2
  • Where do the lines intersect?
  • (2, 0)

2x y 4
x y 2
99
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
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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.
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