today’s agenda e – e – how do you rewrite equations from standard form to slope-intercept...

Today’s AgendaE – E – How do you rewrite equations from standard form to

slope-intercept form?A - A - Warm-up: Writing systems equations

Journal: Describe 2 ways to graph a lineReview: Looking at y=mx+b

T - T - New: Rewriting equations from standard to slope-intercept form (graphic organizer)Classwork: Racing Game

S – S – Journal writing to answer LEQHomework: Rewriting Equations worksheet

104/21/23 Geometry CP

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?

04/21/23 Geometry CP 2

Rewriting Equations Rewriting Equations

LLesson EEssential QQuestion:

How do you rewrite equations from standard form to graphing form?

04/21/23 3Geometry CP

2 Forms of Linear Equations 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: y=mx+b


Important!!! 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


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

Review: -Review: -Writing Equations Writing Equations Given Slope & Y-interceptGiven Slope & Y-intercept


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


Standard Form to Slope Intercept Form

Ax + By = C

to

y = mx + b04/21/23 Geometry CP 8

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.


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


Converting from Converting from standard formstandard form (Ax + By = C) (Ax + By = C) to to slope-intercept formslope-intercept form (y = mx + b) (y = mx + b)


Converting Standard to Slope-Intercept form

2x + 3y = 62x + 3y = 6 ax + by = cax + by = c

-2x-2x -2x-2x 3y = 6 - 2x3y = 6 - 2x

33 33 33

y = 2 -y = 2 - 22

33xx

y = - + 2y = - + 22233

xx y = mx + by = mx + b

WE WANT THIS FORM!!!WE WANT THIS FORM!!!

(Standard Form)(Standard Form)

(Slope- Intercept)(Slope- Intercept)


3 Powerful Moves to get your equation into y=

1. MOVE X2. DROP ALL3. DIVIDE ALL

6X + 4Y = 12


1. MOVE X Add or Subtract the x

term to the other side of equals.

6X + 4Y = 12

6X + 4Y = 12 -6x -6x



2. DROP ALLBring all terms down in

order.

Do not add or subtract unlike terms!!!

6X + 4Y = 12

6X + 4Y = 12 -6x -6x

4y = 12 -6x



3. DIVIDE ALLDivide each term by

the number attached to y

keep slope a fraction!

6X + 4Y = 12

6X + 4Y = 12 -6x -6x

4y = 12 -6x 4 4 4 y= 3 – 3/2

x

2



1. MOVE X2. DROP ALL3. DIVIDE ALL

-4X + 3Y = 12



1. MOVE X Add or Subtract the x

term to the other side of equals.

-4X + 3Y = 12

-4X + 3Y = 12 +4x

+4x



2. DROP ALLBring all terms down in

order.

Do not add or subtract unlike terms!!!

-4X + 3Y = 12

-4X + 3Y = 12 +4x +4x

3y = 12 +4x



3. DIVIDE ALLDivide each term by

the number attached to y

keep slope a fraction!

-4X + 3Y = 12

-4X + 3Y = 12 +4x +4x

3y = 12 +4x 3 3 3 y= 4 + 4/3

x

2



The 3 Power Moves to getting linesinto y = form.

1. 2. 3.

MOVE the x term by Adding/Subtracting!

Drop ALL!

Divide ALL!


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


Example

3x + 2y = 18

The first step is: Subtract 3x from both sides

That would look like:

3x + 2y = 18

-3x -3x

2y = 18 – 3x04/21/23 Geometry CP 23

Example

-7x + 14y = 28

The first step is: Add 7x to both sides


-7x + 14y = 28

+7x +7x

14y = 28 + 7x 04/21/23 Geometry CP 24

Let’s Race!

• As quickly and quietly as possible line up please!

• No hitting, touching, pushing, poking…just get in line!

Ready, Set, GO!!04/21/23 Geometry CP 25

4x + 5y = 10Correct! Subtract 4x from both sides!!

Great Job!

4x + 5y = 10 -4x -4x

5y = 10 – 4x 04/21/23 Geometry CP 26

-6x + 3y = 12Correct! Add 6x to both sides!!

Great Job!

-6x + 3y = 12 +6x +6x

3y = 12 + 6x 04/21/23 Geometry CP 27

9x - y = -8Correct! Subtract 9x from both sides!!

Great Job!

9x - y = -8 -9x -9x

- y = -8 – 9x 04/21/23 Geometry CP 28

10x - 20y = 20Correct! Subtract 10x from both sides!!

Great Job!

10x - 20y = 20 -10x -10x

-20y = 20 – 10x 04/21/23 Geometry CP 29


Great Job!

-11x + 11y = 33 +11x +11x

11y = 33 + 11x 04/21/23 Geometry CP 30


Great Job!

-4x + 2y = 8 +4x +4x

2y = 8 + 4x 04/21/23 Geometry CP 31

-8x - 4y = -16Correct! Add 8x to both sides!!

Great Job!

-8x - 4y = -16 +8x +8x

-4y = -16 + 8x 04/21/23 Geometry CP 32

7x + y = -2Correct! Subtract 7x from both sides!!

Great Job!

7x + y = -2 -7x -7x

y = -2 – 7x 04/21/23 Geometry CP 33


Great Job!

2x + 2y = 10 -2x -2x

2y = 10 – 2x 04/21/23 Geometry CP 34

-5x + 3y = -9Correct! Add 5x to both sides!!

Great Job!

-5x + 3y = -9 +5x +5x

3y = -9 + 5x 04/21/23 Geometry CP 35

-8x - 4y = 24Correct! Add 8x to both sides!!

Great Job!

-8x - 4y = 24 +8x +8x

-4y = 24 + 8x 04/21/23 Geometry CP 36

6x – 12y = -36Correct! Subtract 6x from both sides!!

Great Job!

6x – 12y = -36 -6x -6x

-12y = -36 – 6x 04/21/23 Geometry CP 37

-2x – 2y = -14Correct! Add 2x to both sides!!

Great Job!

-2x – 2y = -14 +2x +2x

-2y = -14 + 2x 04/21/23 Geometry CP 38

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


First StepExample Problem 1

6x + 3y = 9



-10x + 2y = 8



x - 2y = 4



-x + y = -2



-8x – 2y = -2


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


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


Example4x + 2y = 18



4x + 2y = 18 -4x -4x

2y = 18 – 4xThe second step is: Divide everything by 2

2y = 18 – 4x2 2 2

Final Result: y = 9 – 2x04/21/23 Geometry CP 47

Example-14x + 7y = 28



-14x + 7y = 28 +14x +14x

7y = 28 + 14xThe second step is: Divide everything by 7

7y = 28 + 14x7 7 7

Final Result: y = 4 + 2x04/21/23 Geometry CP 48

Example-8x – 2y = -10



-8x – 2y = -10 +8x +8x

-2y = -10 + 8xThe second step is: Divide everything by -2

-2y = -10 + 8x

-2 -2 -2Final Result: y = 5 – 4x


Example12x – 6y = 18



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 + 2x04/21/23 Geometry CP 50

Let’s Race Again!

ReadySet

GO!!04/21/23 Geometry CP 51

4x + 5y = 10First Step?

Correct! Subtract 4x from both sides!!

4x + 5y = 10 -4x -4x

5y = 10 – 4x Second Step?

Correct! Divide everything by 5!!04/21/23 Geometry CP 52


5y = 10 – 4x

y = 2 – 4/5x


-6x + 3y = 12

+6x +6x 3y = 12 + 6x

Second Step?



3y = 12 + 6x

y = 4 +2x

9x - y = -8Correct! Subtract 9x from both sides!!

9x - y = -8 -9x -9x - y = -8 – 9x

What’s in front of the y…that is always there…we just don’t write it (because mathematicians are lazy )?

- 1y = -8 – 9x Second Step?

Correct! Divide everything by -1!!04/21/23 Geometry CP 56


-y = -8 – 9x

y = 8 +9x

10x - 20y = 20Correct! Subtract 10x from both sides!!

10x - 20y = 20 -10x -10x -20y = 20 – 10x

Second Step?

Correct! Divide everything by -2004/21/23 Geometry CP 58


-20y = 20 – 10x

y = -1 +1/2x


-11x + 11y = 33 +11x +11x 11y = 33 + 11x

Second Step?

Correct! Divide everything by 11



11y = 33 + 11x

y = 3 +x


-4x + 2y = 8 +4x +4x 2y = 8 + 4x

Second Step?

Correct! Divide everything by 2



2y = 8 + 4x

y = 4 +2x

-8x - 4y = -16Correct! Add 8x to both sides!!

-8x - 4y = -16 +8x +8x -4y = -16 + 8x

Second Step?

Correct! Divide everything by -4!!04/21/23 Geometry CP 64


-4y = -16 + 8x

y = 4 - 2x

7x + y = -2Correct! Subtract 7x from both sides!!

7x + y = -2 -7x -7x

y = -2 – 7x Second Step?

Correct! There is no second step!

It’s already solved for y 04/21/23 Geometry CP 66


y = -2 - 7x

y = -2 -7x


2x + 2y = 10 -2x -2x 2y = 10 – 2x

Second Step?



2y = 10 – 2x

y = 5 -x

-5x + 3y = -9Correct! Add 5x to both sides!!

-5x + 3y = -9 +5x +5x

3y = -9 + 5x Second Step?

Correct! Divide everything by 304/21/23 Geometry CP 70


3y = -9 + 5x

y = -3 +5/3x

-8x - 4y = 24Correct! Add 8x to both sides!!

-8x - 4y = 24 +8x +8x -4y = 24 + 8x

Second Step?



-4y = 24 + 8x

y = -6 - 2x

6x – 12y = -36Correct! Subtract 6x from both sides!!

6x – 12y = -36 -6x -6x

-12y = -36 – 6x Second Step?



-12y = -36 – 6x

y = 3 +1/2x

-2x – 2y = -14Correct! Add 2x to both sides!!

-2x – 2y = -14 +2x +2x

-2y = -14 + 2x Second Step?



-2y = -14 + 2x

y = 7 - x

Putting it all TogetherFirst & Second Step Example Problem 1

35x + 7y = 49



-20x – 5y = -30



-6x + 3y = 24



-x + 2y = 4



x + y = 8



x + 4y = 8


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


ERROR ANALYSISERROR ANALYSIS

JARED12x + 3y = 9 3y = 9 – 12x y = 3 – 4x

Ali12x + 3y = 9 4x + y = 3 y = -4x + 3

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.

Molly12x + 3y = 9 3y = 9 – 12x y = 3 – 12x

Mia12x + 3y = 9 3y = 9 – 12x y = 3 – 4x y = 4x - 304/21/23 Geometry CP 85

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!


HOMEWORK!

Complete the Slope Intercept and Standard Form wsht


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 OTHER’S WORK.

• This will be a graded assignment to earn bonus points on your quiz


You have 2 minutes to find your partner!


Purple GroupColette

DanKaylaSydni

JonathanPhil

Pink GroupMeganTyheimTiyanaAlisa

Courtney

Orange GroupDaysiaTaylor

Chris MShielaAndy

Chris NAshleySteven

Purple Group

Directions: • For the following problems

find the x & y intercepts. Show work!

• Don’t forget that the x intercept happens when y=0 and the y intercept happens when x=0

• Write all intercepts as an ordered pair (x,y)

Finding X & Y intercepts

a. 2x – 3y = 12

b. 2x + 3y = 12

c. 3x – y = 6

d. y – x = 5


Orange Group

Directions:

• Rewrite each equation into slope-intercept form (y =mx+b)

• Identify the slope and y-intercept

• Show all work!

Rewriting Equations

a) 3x + 2y = 28

b) 5y = 15 – 2x

c) 3y + 9 = 2x


Pink Group

Directions:• Rewrite each equation

into slope-intercept form (y =mx+b)

• Identify the slope and y-intercept

• Don’t forget the 3 POWER steps..Use your notes if needed!

• Show all work!

Rewriting Equations

a) x + y = 20

b) 5x + 4y = 24

c) 3x – 2y = 12



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Solve Systems of Solve Systems of Equations by the Graphing Equations by the Graphing

MethodMethod

LLesson EEssential QQuestion:

Describe the types of solutions a system of equations can have?


What is a system of What is a system of equations?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.


Intersecting Lines Intersecting Lines (One Solution)(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)


Find the solution to the following Find the solution to the following system using the Graphing Methodsystem 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 + 4y –int. = (0, 4) and

Slope = -2/1 or 2/-1

y = x - 2y – int. = (0, -2) and Slope = 1/1 or -1/-1


Step 2: Graph the Step 2: Graph the equations.equations.

y = -2x + 4

y = x - 2

Where do the lines intersect?(2, 0)

2x + y = 4

x – y = 2


Step 3: Check your answer!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) + 40 = -4 + 40 = 0

y = -x + 2(0) = -(2) + 20 = 0

Nice job…let’s look how to solve it usingthe graphing calculator!


Geometry CP 100

Quick Stop & Jot

DO ALL LINES ALWAYS HAVE A POINT OF INTERSECTION?

WHAT OTHER TYPES OF SOLUTIONS CAN SYSTEMS OF EQUATIONS HAVE?

04/21/23

Geometry CP 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?04/21/23

Parallel LinesParallel Lines(No Solution)(No Solution)

These lines never intersect! Since the lines never cross,

there is NO SOLUTIONNO SOLUTION!

Parallel lines have the same slope with different y-intercepts.

2Slope = = 2

1y-intercept = 2

y-intercept = -1


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

y = 2x – 3

y = 2x + 1

Graph both equations using slope and y-intercept.


Step 2: Graph the Step 2: Graph the equations.equations.

y = 2x – 3m = 2 and b = -3

y = 2x + 1m = 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 don’t

have to graph them!


Step 3: Check your answer!Step 3: Check your answer!

Not a lot to check…Just make sure you set up your equations correctly.

I double-checked it and I did it right…


Geometry Honors 106

Another type of solutionAnother 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?

04/21/23

Infinitely Many SolutionsInfinitely Many Solutions

SAME LINE

Coinciding Lines Coinciding Lines (Infinitely Many Solutions)(Infinitely Many Solutions)

These lines are the same! Since the lines are on top of

each other, there are INFINITELY MANY INFINITELY MANY SOLUTIONSSOLUTIONS!

Coinciding lines have the same slope and y-intercepts.

2Slope = = 2

1y-intercept = -1


Geometry CP 109

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

Graph 6x + 4y = 12 and 3x + 2y = 6

04/21/23

Geometry CP 110

JOURNAL: Does it have a solution?

1) 1)

Determine whether the following have one, none, or infinite Determine whether the following have one, none, or infinite solutions by looking at the solutions by looking at the slopeslope and and y-intercepty-intercept. . Explain your reasoning. Explain your reasoning.

y = 4 -1/2 xy = 4 -1/2 x

y = 2x + 4y = 2x + 4

3) 3) 2) 2) y = -3/4x + 6y = -3/4x + 6

y = -3/4x - 6y = -3/4x - 6

y = -6x + 8y = -6x + 8

y + 6x = 8y + 6x = 8

04/21/23

Geometry CP 111

Does it have a solution?

1) 1)

Determine whether the following have one, none, or infinite Determine whether the following have one, none, or infinite solutions by just looking at the solutions by just looking at the slopeslope and and y-intercepts.y-intercepts.

3) 3) 2) 2)

ANS:ANS: One SolutionOne Solution

ANS:ANS: No SolutionNo Solution

ANS:ANS: Infinite SolutionsInfinite Solutions

04/21/23

y = -3/4x + 6y = -3/4x + 6

y = -3/4x - 6y = -3/4x - 6

y = -6x + 8y = -6x + 8

y + 6x = 8y + 6x = 8

y = 4 -1/2 xy = 4 -1/2 x

y = 2x + 4y = 2x + 4

What is the solution of the What is the solution of the system graphed below?system graphed below?

1. (2, -2)

2. (-2, 2)

3. No solution

4. Infinitely many solutions


What is the solution of this What is the solution of this system using the Graphing system using the Graphing

Method?Method?

y = 2x - 2y = 2x + 1

1. (2, -2)

2. (2, 1)

3. No solution




Method?Method?

y = 2x - 2y = 1/2x + 4

1. (4, 6)

2. (6, 4)

3. No solution




Method?Method?

y = 3x - 8y = 3x - 8

1. (3, 1)

2. (4, 4)

3. No solution




Method?Method?

y = 4x - 2-4x + y = -2

1. (4, -2)

2. (-2, 4)

3. No solution



Solving a system of equations by Solving a system of equations by the the Graphing Method Graphing Method

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

Step 1: Graph both equations.

Step 2: Do the graphs intersect?

Step 3: Check your solution.

Graph using slope and y – intercept. Be sure to use a ruler and graph paper!

This is the solution! LABEL the solution (x, y)!

Substitute the x and y values into both equations to verify the point is a solution to both equations.


Summarize TimeSummarize Time

In your journals, write today’s 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.


Geometry CP 119

GRAPHING CALCULATOR

Rewrite equation in y = formUse the INTERSECT function to find the

intersection point

04/21/23

Geometry CP 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

04/21/23

Geometry CP 122

GRAPHING CALCULATOR 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

04/21/23

(2, - 6)

(3, -2)

(-2, 7)

(-2, 5)

No solution

Infinitely Many

““All I Do Is Solve” (Part I)All I Do Is Solve” (Part I)

http://www.youtube.com/watch?v=qxHCEwrpMw0&NR=1


Check Your Check Your UnderstandingUnderstanding

Solve the system of equations using the Graphing Method. Check your solution.

y = 3x – 3

y = -x + 1


Group Self-Evaluation Group Self-Evaluation FormForm

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


Homework AssignmentHomework Assignment

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


today’s agenda e – e – how do you rewrite equations from standard form to slope-intercept...

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