Lesson Essential Question: Rewriting Equations  Lesson Essential Question: How do you rewrite equations from standard form to graphing form? Write LEQ in journal 12/8/2018 Geometry CP

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 Write 2 forms of linear equations in journal 12/8/2018 Geometry CP

Important!!!  This is one of the BIG concepts you will need for high school. Slope – Intercept Form y = mx + b m represents the slope b represents the y-intercept Write in journal 12/8/2018 Geometry CP

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 12/8/2018 Geometry CP

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 + b

Converting from standard form (Ax + By = C) to slope-intercept form (y = mx + b) Tape in journal

Converting Standard to Slope-Intercept form 2x + 3y = 6 ax + by = c (Standard Form) -2x -2x WE WANT THIS FORM!!! 3y = 6 - 2x 3 y = 2 - 2 3 x y = - + 2 2 3 x y = mx + b (Slope- Intercept)

3 Powerful Moves to get your equation into y= 6X + 4Y = 12 MOVE X DROP ALL DIVIDE ALL 12/8/2018 Geometry CP

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. 12/8/2018 Geometry CP

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!!! 12/8/2018 Geometry CP

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! 12/8/2018 Geometry CP

3 Powerful Moves to get your equation into y= -4X + 3Y = 12 MOVE X DROP ALL DIVIDE ALL 12/8/2018 Geometry CP

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. 12/8/2018 Geometry CP

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!!! 12/8/2018 Geometry CP

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! 12/8/2018 Geometry CP

The 3 Power Moves to getting lines into y = form. 1. 2. 3. MOVE the x term by Adding/Subtracting! Write in journal Drop ALL! Divide ALL! 12/8/2018 Geometry CP

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  12/8/2018 Geometry CP

Example 3x + 2y = 18 That would look like: 3x + 2y = 18 -3x -3x The first step is: Subtract 3x from both sides That would look like: 3x + 2y = 18 -3x -3x 2y = 18 – 3x 12/8/2018 Geometry CP

Example -7x + 14y = 28 That would look like: -7x + 14y = 28 +7x +7x The first step is: Add 7x to both sides That would look like: -7x + 14y = 28 +7x +7x 14y = 28 + 7x 12/8/2018 Geometry CP

Let’s Race! As quickly and quietly as possible line up please! No hitting, touching, pushing, poking…just get in line! Ready, Set, GO!! 12/8/2018 Geometry CP

Correct! Subtract 4x from both sides!! 4x + 5y = 10 Correct! Subtract 4x from both sides!! Great Job! 4x + 5y = 10 -4x -4x 5y = 10 – 4x 12/8/2018 Geometry CP

Correct! Add 6x to both sides!! -6x + 3y = 12 Correct! Add 6x to both sides!! Great Job! -6x + 3y = 12 +6x +6x 3y = 12 + 6x 12/8/2018 Geometry CP

Correct! Subtract 9x from both sides!! 9x - y = -8 Correct! Subtract 9x from both sides!! Great Job! 9x - y = -8 -9x -9x - y = -8 – 9x 12/8/2018 Geometry CP

Correct! Subtract 10x from both sides!! 10x - 20y = 20 Correct! Subtract 10x from both sides!! Great Job! 10x - 20y = 20 -10x -10x -20y = 20 – 10x 12/8/2018 Geometry CP

Correct! Add 11x to both sides!! -11x + 11y = 33 Correct! Add 11x to both sides!! Great Job! -11x + 11y = 33 +11x +11x 11y = 33 + 11x 12/8/2018 Geometry CP

Correct! Add 4x to both sides!! -4x + 2y = 8 Correct! Add 4x to both sides!! Great Job! -4x + 2y = 8 +4x +4x 2y = 8 + 4x 12/8/2018 Geometry CP

Correct! Add 8x to both sides!! -8x - 4y = -16 Correct! Add 8x to both sides!! Great Job! -8x - 4y = -16 +8x +8x -4y = -16 + 8x 12/8/2018 Geometry CP

Correct! Subtract 7x from both sides!! 7x + y = -2 Correct! Subtract 7x from both sides!! Great Job! 7x + y = -2 -7x -7x y = -2 – 7x 12/8/2018 Geometry CP

Correct! Subtract 2x from both sides!! 2x + 2y = 10 Correct! Subtract 2x from both sides!! Great Job! 2x + 2y = 10 -2x -2x 2y = 10 – 2x 12/8/2018 Geometry CP

Correct! Add 5x to both sides!! -5x + 3y = -9 Correct! Add 5x to both sides!! Great Job! -5x + 3y = -9 +5x +5x 3y = -9 + 5x 12/8/2018 Geometry CP

Correct! Add 8x to both sides!! -8x - 4y = 24 Correct! Add 8x to both sides!! Great Job! -8x - 4y = 24 +8x +8x -4y = 24 + 8x 12/8/2018 Geometry CP

Correct! Subtract 6x from both sides!! 6x – 12y = -36 Correct! Subtract 6x from both sides!! Great Job! 6x – 12y = -36 -6x -6x -12y = -36 – 6x 12/8/2018 Geometry CP

Correct! Add 2x to both sides!! -2x – 2y = -14 Correct! Add 2x to both sides!! Great Job! -2x – 2y = -14 +2x +2x -2y = -14 + 2x 12/8/2018 Geometry CP

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  12/8/2018 Geometry CP

First Step Example Problem 1 6x + 3y = 9 12/8/2018 Geometry CP

First Step Example Problem 2 -10x + 2y = 8 12/8/2018 Geometry CP

First Step Example Problem 3 x - 2y = 4 12/8/2018 Geometry CP

First Step Example Problem 4 -x + y = -2 12/8/2018 Geometry CP

First Step Example Problem 5 -8x – 2y = -2 12/8/2018 Geometry CP

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  While students are working on the 10 problems – write the problems on the board so that the students can come up and put their work on the board. All 10 on the board at once. With those 10 – show the process of the second step. Have the students just watch and help as you solve them. They should be watching – not writing anything down. 12/8/2018 Geometry CP

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  12/8/2018 Geometry CP

Example 4x + 2y = 18 Final Result: y = 9 – 2x That would look like: 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 12/8/2018 Geometry CP

Example -14x + 7y = 28 Final Result: y = 4 + 2x That would look like: 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 12/8/2018 Geometry CP

Example -8x – 2y = -10 Final Result: y = 5 – 4x That would look like: 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 12/8/2018 Geometry CP

Example 12x – 6y = 18 Final Result: y = -3 + 2x That would look like: 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 12/8/2018 Geometry CP

Let’s Race Again! ReadySet GO!! 12/8/2018 Geometry CP

4x + 5y = 10 4x + 5y = 10 -4x -4x 5y = 10 – 4x First Step? Correct! Subtract 4x from both sides!! 4x + 5y = 10 -4x -4x 5y = 10 – 4x Second Step? Correct! Divide everything by 5!! 12/8/2018 Geometry CP

5y = 10 – 4x y = 2 – 4/5x 12/8/2018 Geometry CP

Correct! Add 6x to both sides!! Correct! Divide everything by 3!! -6x + 3y = 12 Correct! Add 6x to both sides!! -6x + 3y = 12 +6x +6x 3y = 12 + 6x Second Step? Correct! Divide everything by 3!! 12/8/2018 Geometry CP

3y = 12 + 6x y = 4 +2x 12/8/2018 Geometry CP

9x - y = -8 9x - y = -8 -9x -9x - y = -8 – 9x - 1y = -8 – 9x Correct! 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!! 12/8/2018 Geometry CP

-y = -8 – 9x y = 8 +9x 12/8/2018 Geometry CP

10x - 20y = 20 Correct! Subtract 10x from both sides!! 10x - 20y = 20 -10x -10x -20y = 20 – 10x Second Step? Correct! Divide everything by -20 12/8/2018 Geometry CP

-20y = 20 – 10x y = -1 +1/2x 12/8/2018 Geometry CP

Correct! Add 11x to both sides!! Correct! Divide everything by 11 -11x + 11y = 33 Correct! Add 11x to both sides!! -11x + 11y = 33 +11x +11x 11y = 33 + 11x Second Step? Correct! Divide everything by 11 12/8/2018 Geometry CP

11y = 33 + 11x y = 3 +x 12/8/2018 Geometry CP

Correct! Add 4x to both sides!! Correct! Divide everything by 2 -4x + 2y = 8 Correct! Add 4x to both sides!! -4x + 2y = 8 +4x +4x 2y = 8 + 4x Second Step? Correct! Divide everything by 2 12/8/2018 Geometry CP

2y = 8 + 4x y = 4 +2x 12/8/2018 Geometry CP

Correct! Add 8x to both sides!! Correct! Divide everything by -4!! -8x - 4y = -16 Correct! Add 8x to both sides!! -8x - 4y = -16 +8x +8x -4y = -16 + 8x Second Step? Correct! Divide everything by -4!! 12/8/2018 Geometry CP

-4y = -16 + 8x y = 4 - 2x 12/8/2018 Geometry CP

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! It’s already solved for y  12/8/2018 Geometry CP

y = -2 - 7x y = -2 -7x 12/8/2018 Geometry CP

2x + 2y = 10 Correct! Subtract 2x from both sides!! 2x + 2y = 10 -2x -2x 2y = 10 – 2x Second Step? Correct! Divide everything by 2!! 12/8/2018 Geometry CP

2y = 10 – 2x y = 5 -x 12/8/2018 Geometry CP

Correct! Add 5x to both sides!! Correct! Divide everything by 3 -5x + 3y = -9 Correct! Add 5x to both sides!! -5x + 3y = -9 +5x +5x 3y = -9 + 5x Second Step? Correct! Divide everything by 3 12/8/2018 Geometry CP

3y = -9 + 5x y = -3 +5/3x 12/8/2018 Geometry CP

Correct! Add 8x to both sides!! Correct! Divide everything by -4 -8x - 4y = 24 Correct! Add 8x to both sides!! -8x - 4y = 24 +8x +8x -4y = 24 + 8x Second Step? Correct! Divide everything by -4 12/8/2018 Geometry CP

-4y = 24 + 8x y = -6 - 2x 12/8/2018 Geometry CP

6x – 12y = -36 6x – 12y = -36 -6x -6x -12y = -36 – 6x Correct! Subtract 6x from both sides!! 6x – 12y = -36 -6x -6x -12y = -36 – 6x Second Step? Correct! Divide everything by -12 12/8/2018 Geometry CP

-12y = -36 – 6x y = 3 +1/2x 12/8/2018 Geometry CP

Correct! Add 2x to both sides!! Correct! Divide everything by -2 -2x – 2y = -14 Correct! Add 2x to both sides!! -2x – 2y = -14 +2x +2x -2y = -14 + 2x Second Step? Correct! Divide everything by -2 12/8/2018 Geometry CP

-2y = -14 + 2x y = 7 - x 12/8/2018 Geometry CP

Putting it all Together First & Second Step Example Problem 1 35x + 7y = 49 12/8/2018 Geometry CP

Putting it all Together First & Second Step Example Problem 2 -20x – 5y = -30 12/8/2018 Geometry CP

Putting it all Together First & Second Step Example Problem 3 -6x + 3y = 24 12/8/2018 Geometry CP

Putting it all Together First & Second Step Example Problem 4 -x + 2y = 4 12/8/2018 Geometry CP

Putting it all Together First & Second Step Example Problem 5 x + y = 8 12/8/2018 Geometry CP

Putting it all Together First & Second Step Example Problem 6 x + 4y = 8 12/8/2018 Geometry CP

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  While students are working on the 10 problems – write the problems on the board so that the students can come up and put their work on the board. All 10 on the board at once. With those 10 – show the process of the second step. Have the students just watch and help as you solve them. They should be watching – not writing anything down. 12/8/2018 Geometry CP

ERROR ANALYSIS Molly JARED 12x + 3y = 9 12x + 3y = 9 3y = 9 – 12x 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 12/8/2018 Geometry CP

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! 12/8/2018 Geometry CP

Purple Group Directions: Finding X & Y intercepts 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) a. 2x – 3y = 12 b. 2x + 3y = 12 c. 3x – y = 6 d. y – x = 5 12/8/2018 Geometry CP

Orange Group Directions: Rewriting Equations Rewrite each equation into slope-intercept form (y =mx+b) Identify the slope and y-intercept Show all work! 3x + 2y = 28 b) 5y = 15 – 2x c) 3y + 9 = 2x 12/8/2018 Geometry CP

Pink Group Directions: Rewriting Equations a) x + y = 20 5x + 4y = 24 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! a) x + y = 20 5x + 4y = 24 3x – 2y = 12 12/8/2018 Geometry CP

Solve Systems of Equations by the Graphing Method Lesson Essential Question: Describe the types of solutions a system of equations can have? 12/8/2018 Geometry CP

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. 12/8/2018 Geometry CP

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) 12/8/2018 Geometry CP

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 12/8/2018 Geometry CP

Step 2: Graph the equations. y = -2x + 4 y = x - 2 Where do the lines intersect? (2, 0) 2x + y = 4 x – y = 2 12/8/2018 Geometry CP

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 Nice job…let’s look how to solve it using the graphing calculator! 12/8/2018 Geometry CP

Quick Stop & Jot DO ALL LINES ALWAYS HAVE A POINT OF INTERSECTION? WHAT OTHER TYPES OF SOLUTIONS CAN SYSTEMS OF EQUATIONS HAVE? 12/8/2018 Geometry CP

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? 12/8/2018 Geometry CP

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. 12/8/2018 Geometry CP

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. 12/8/2018 Geometry CP

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 don’t have to graph them! 12/8/2018 Geometry CP

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… 12/8/2018 Geometry CP

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? 12/8/2018 Geometry Honors

Infinitely Many Solutions SAME LINE

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. 12/8/2018 Geometry CP

Find the solution to the following system by the Graphing Method Graph 6x + 4y = 12 and 3x + 2y = 6 12/8/2018 Geometry CP

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) y = 4 -1/2 x y = 2x + 4 y = -6x + 8 y + 6x = 8 3) y = -3/4x + 6 y = -3/4x - 6 12/8/2018 Geometry CP

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) y = 4 -1/2 x y = 2x + 4 2) y = -6x + 8 y + 6x = 8 3) y = -3/4x + 6 y = -3/4x - 6 ANS: One Solution ANS: Infinite Solutions ANS: No Solution 12/8/2018 Geometry CP

What is the solution of the system graphed below? (2, -2) (-2, 2) No solution Infinitely many solutions 12/8/2018 Geometry CP

What is the solution of this system using the Graphing Method? y = 2x - 2 y = 2x + 1 (2, -2) (2, 1) No solution Infinitely many solutions 12/8/2018 Geometry CP

What is the solution of this system using the Graphing Method? y = 2x - 2 y = 1/2x + 4 (4, 6) (6, 4) No solution Infinitely many solutions 12/8/2018 Geometry CP

What is the solution of this system using the Graphing Method? y = 3x - 8 (3, 1) (4, 4) No solution Infinitely many solutions 12/8/2018 Geometry CP

What is the solution of this system using the Graphing Method? y = 4x - 2 -4x + y = -2 (4, -2) (-2, 4) No solution Infinitely many solutions 12/8/2018 Geometry CP

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. 12/8/2018 Geometry CP

“All I Do Is Solve” (Part I) http://www.youtube.com/watch?v=qxHCEwrpMw0&NR=1 12/8/2018 Geometry CP

Check Your Understanding Solve the system of equations using the Graphing Method. Check your solution. y = 3x – 3 y = -x + 1 12/8/2018 Geometry CP