2. Today’s Agenda E -E - How can graphs be used to solve systems of linear equations? A -A - Warm-up: Bell Ringer Patterns B Review: Graph linear equations from slope-intercept form (graphic organizer) T -T - New: System of Linear Equations (graphic organizer and video) –Class work: A Good Deal Shoulder Buddy Practice Problem S – 3-2-1S – 3-2-1 Homework: Graphing Equations worksheet #1-6 19/6/2015Smyrna High School Geometry CP

3. Warm-up Bell Ringer Day 1~Discovering Patterns… Complete the pattern with your group. Then answer the questions as best you can. 9/6/2015Smyrna High School Geometry CP 2

4. Pattern B Draw the penny pattern in figure 4. How many pennies are in figure 5? How many pennies are in figure 10? What figure has 63 pennies? Write a rule for this pattern? 9/6/2015Smyrna High School Geometry CP 3

5. Unit 1 – Lesson 3: Systems of Linear Equations Systems of Equations Two or more linear equations that have to be solved simultaneously. 9/6/20154Smyrna High School Geometry CP

6. How to Graph an Equation in Slope-Intercept Form y = mx + b In the slope-intercept form of a linear equation, each letter represents a special value. m = the slope of the equation (steepness). b = the y-intercept (starting number). With these two facts, an equation in slope- intercept form can be graphed. Check Skills You Will Need 59/6/2015Smyrna High School Geometry CP

7. How to Graph an Equation in Slope-Intercept Form y = mx + b With these two facts, lets look at some examples. y = 4x + 2 First locate point b on the graph and mark it. Since this is the point where the graph of the equation crosses the y-axis, the x value will be zero and the y value will be b in the equation (0, b). Check Skills You Will Need 9/6/20156Smyrna High School Geometry CP

8. How to Graph an Equation in Slope-Intercept Form y = mx + b Second, determine the slope (m) of the equation. What number is before the x in the equation? If there is not a number in front of the letter x, remember the number 1 is understood. Check Skills You Will Need 9/6/20157Smyrna High School Geometry CP

9. How to Graph an Equation in Slope-Intercept Form y = mx + b Slope is commonly know as the rise/run. The slope of the example can be written as 4/1. Use these values to plot some more points. Check Skills You Will Need 9/6/20158Smyrna High School Geometry CP

10. How to Graph an Equation in Slope-Intercept Form y = mx + b Start at the y-intercept (0, 2). From this point use the slope of the equation to find a second point. The slope of the line is 4/1 so count up 4 spaces (the rise) and count to the right 1 space (the run). Repeat these steps to find a third point. Check Skills You Will Need 99/6/2015Smyrna High School Geometry CP

11. How to Graph an Equation in Slope-Intercept Form y = mx + b Remember, the slope of the line will determine the direction of the rise and run. Going up is a positive direction. Then run to the right. Going down is a negative direction. Then run to the right. Check Skills You Will Need 9/6/201510Smyrna High School Geometry CP

12. How to Graph an Equation in Slope-Intercept Form y = mx + b Remember, a negative divided by a negative is a positive. Positive slopes will start lower on the left side of the graph and end up higher on the right side of the graph. Negative slopes will start higher on the left side of the graph and end up lower on the right side of the graph. Check Skills You Will Need 119/6/2015Smyrna High School Geometry CP

13. Pattern C How many toothpicks are in figure 5? How many toothpicks are in figure 10? What figure has 44 pennies? Write a rule for this pattern. 9/6/2015Smyrna High School Geometry CP 12

14. How to Graph an Equation in Slope-Intercept Form y = mx + b For practice, graph the following equations in slope- intercept form. 1.y = -x + 6 2.y = 2x – 4 3.y = ¼ x Check Skills You Will Need 9/6/201513Smyrna High School Geometry CP

15. 9/6/2015Smyrna High School Geometry CP 14 (0, 6) -1/1 (0, - 4) 2/1 (0, 0)1/4

16. Graph! http://www.youtube.com/watch?v=2BHzXIt kByUhttp://www.youtube.com/watch?v=2BHzXIt kByU 9/6/2015Smyrna High School Geometry CP 15

17. Ticket In The Door Recap For your equation do the following: a)Circle & Identify the y-intercept (don’t forget to circle the sign also) b)Graph the y-intercept ( always on the y-axis ) c)Identify the slope (write it as a fraction) d)Use slope to plot more points (ryse/run) e)Draw a line through the points You may use your “yellow” graphic organizer to help guide you through the steps! 9/6/2015Smyrna High School Geometry CP 16

18. 9/6/2015Smyrna High School Geometry CP 17

19. Homework check Use the red pen to make corrections to your homework Now, is NOT the time to do your homework I should not see any writing utensils except the red pen If you didn’t do your homework then you can turn it in tomorrow for half credit kuta graphing lines in slope intercept form hw.pdf 9/6/2015Smyrna High School Geometry CP 18

20. Systems of Equations Let’s take a look at a few examples. I Notice, I WonderYour task: I Notice, I Wonder. NOTICE –Discuss and write down some things that you NOTICE about the examples. WONDER –Discuss and write down some things that you WONDER about the examples. 9/6/201519Smyrna High School Geometry CP

21. Systems of Equations: Example 1 9/6/201520 Smyr na High Scho ol Geo metry CP

22. Systems of Equations: Example 2 9/6/201521Smyrna High School Geometry CP

23. Systems of Equations: Example 3 9/6/201522Smyrna High School Geometry CP

24. 3 Methods to Solve System of Equations Graphing Method Substitution Method Elimination Method 9/6/201523Smyrna High School Geometry CP

25. 3 Possible Solution Types 1 Exact Solution No Solution Infinitely Many Solutions 9/6/201524Smyrna High School Geometry CP

26. 1 Exact Solution 9/6/201525Smyrna High School Geometry CP Point of intersection

27. Smyrna High School Geometry CP 26 Another type of solution How would you describe these lines? Y = 3x + 2 Y = 3x - 4 What do you think the solution (point of intersection) is? 9/6/2015

28. No Solution Parallel Lines Will not intersect same slope different y- intercepts 9/6/2015Smyrna High School Geometry CP 27

29. Smyrna High School Geometry CP 28 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 (point of intersection) is? 9/6/2015

30. Infinitely Many Solutions SAME LINE They are the SAME equation (just written in different forms) They have same slope and same y-intercept. 9/6/2015Smyrna High School Geometry CP 29

31. The Goal of Solving Systems To find one pair (x, y) of values that satisfies both linear equations. –The one pair of values that makes both equations true. 9/6/201530Smyrna High School Geometry CP

32. Hamilton High School 9/6/201531Smyrna High School Geometry CP

33. Hamilton High School Students in the Hamilton High School science club want to raise $240 to buy a special eyepiece for the high- powered telescope at their school. The PTA offered to pay club members for an after-school work project that would clean up a nearby park and recreation center building. Because the outdoor work was harder and dirtier, the deal with the PTA would pay $20 for each outdoor worker ad $10 for each indoor worker. The club had 18 members eager to work on the project. 9/6/2015Smyrna High School Geometry CP 32

34. Hamilton High School What do the variables x and y represent in these equations? What problem condition is represented by the equation 20x + 10y = 240? What problem condition is represented by the equation x + y = 18? Determine 3 combinations of numbers of outdoor and indoor workers that will allow the club to earn just enough money to buy the telescope eyepiece? 9/6/2015Smyrna High School Geometry CP 33

35. Think-Pair-Share Let’s Look at the Solution Complete Problem 1 - Parts a, b, c & d. Try it on your own Think : 5 minutes Check with a partner Pair : 2 minutes Whole group discussion Share : 5 minutes 9/6/201534Smyrna High School Geometry CP

36. Hamilton High School What does the x represent? x: # of outdoor workers What does the y represent? y: # of indoor workers 9/6/2015Smyrna High School Geometry CP 35

37. Hamilton High School 20x + 10y = 240 What does this equation represent in the problem? 20x + 10y = 240 shows the amount of money that can be earned depending on the # of outdoor and indoor workers 9/6/2015Smyrna High School Geometry CP 36

38. Hamilton High School x + y = 18 x + y = 18 What does this equation represent in the problem? x + y = 18 shows that the # of club members who will work 9/6/2015Smyrna High School Geometry CP 37

39. Let’s finish the problem Now, take a look at your 3 combination of workers from part d. Will any of the combinations also put each willing club member to work? 9/6/2015Smyrna High School Geometry CP 38

40. 9/6/2015Smyrna High School Geometry CP 39

41. Today’s Agenda E -E - How can graphs be used to solve systems of linear equations? A -A - Warm-up: Bell Ringer Patterns Review: Graph linear equations from slope-intercept form (graphic organizer) T -T - New: Graph linear equations from intercepts (graphic organizer) –Class work: A Good Deal Shoulder Buddy Practice Problem, Verifying solutions, Solving systems S – Journal entryS – Journal entry Homework: Pattern D 409/6/2015Smyrna High School Geometry CP

42. WARMUP: Pattern E Draw the pattern for figure 5 How many white tiles are in figure 5. How many white tiles are in figure 8? What figure has 72 white tiles. Write a rule for the pattern. 9/6/2015Smyrna High School Geometry CP 41

43. HOMEWORK: Pattern D How many toothpicks are in figure 5? How many toothpicks are in figure 10? What figure has 44 pennies? Write a rule for this pattern. 9/6/2015Smyrna High School Geometry CP 42

44. How Do We Graph a Linear Equation??? One way to graph a linear equation is in the form y = mx + b, where m is the slope and b is the y-intercept 9/6/201543Smyrna High School Geometry CP

45. How Do We Graph a Linear Equation not in y=mx+b form??? Let’s Practice: -3x + 2y = 12 Since it is not in y = mx + b, we will find the x- and y-intercepts and graph those points. Find the x-intercept: Let y = 0 and solve the equation for x. Find the y-intercept: Let x = 0 and solve the equation for y. 9/6/201544Smyrna High School Geometry CP

46. How Do We Graph a Linear Equation??? Let’s Practice: -3x + 2y = 12 Find the x-intercept: Let y = 0 and solve the equation for x. -3x + 2(0) = 12 -3x = 12 -3 -3 x = -4 or (-4, 0) 9/6/201545Smyrna High School Geometry CP

47. How Do We Graph a Linear Equation??? Let’s Practice: -3x + 2y = 12 Find the y-intercept: Let x = 0 and solve the equation for y. -3(0) + 2y = 12 2y= 12 2 2 y = 6 or (0, 6) 9/6/201546Smyrna High School Geometry CP

48. How Do We Graph a Linear Equation??? You try… 6x - 4y = 12 9/6/201547Smyrna High School Geometry CP

49. Time to practice… Let’s do #5 & 6 together Now it is your turn to finish the odd problems (1, 3, 7) Make sure that you have me check these problems (10 mins) If you finish early, go ahead and finish the even problems (2, 4, 8). If you don’t finish the evens in class, you can finish them for homework. graphing lines using intercepts.pdf 9/6/2015Smyrna High School Geometry CP 48

50. How Do We Graph a Linear Equation??? Let’s Practice with Hamilton High: 20x + 10y = 240 Find the x-intercept: Let y = 0 and solve the equation for x. 20x + 10(0) = 240 20x = 240 20 x = 12 or (12, 0) 9/6/201549Smyrna High School Geometry CP

51. How Do We Graph a Linear Equation??? Let’s Practice: 20x + 10y = 240 Find the y-intercept: Let x = 0 and solve the equation for y. 20(0) + 10y = 240 10x = 240 10 y = 24 or (0, 24) 9/6/201550Smyrna High School Geometry CP

52. How Do We Graph a Linear Equation??? Let’s Practice: x + y = 18 9/6/201551Smyrna High School Geometry CP Find the x-intercept: Let y = 0 and solve the equation for x. x + (0) = 18 x = 18 x = 18 or (18, 0)

53. How Do We Graph a Linear Equation??? Let’s Practice: x + y = 18 9/6/201552Smyrna High School Geometry CP Find the y-intercept: Let x = 0 and solve the equation for y. (0) + y = 18 y = 18 y = 18 or (0, 18)

54. Putting It All Together ! Complete part f from the Hamilton High problem. Plot your intercepts to graph both lines. What is your solution to the system? 9/6/2015Smyrna High School Geometry CP 53

55. Let’s verify How can we verify that (6, 12) is the solution to the system of equation: 20x + 10y = 240 x + y = 18 You must verify the solution into BOTH equations for x AND y. 20x + 10y = 240 20(6) + 10(12) = 240 120 + 120 = 240 240 = 240 x + y = 18 6 + 12 = 18 18 = 18 Smyrna High School Geometry CP 9/6/201554

56. How do you verify solutions? (Is it the answer?) 9/6/2015 Smyrna High School Geometry CP 55 Determine whether the point (3,8) is a solution to each system of equations. 2x + y = 14 x + y = 11 y = -x – 5 y = x + 5 4x – y = - 4 3x – 2y = 7

57. Verifying Solutions 9/6/2015 Smyrna High School Geometry CP 56 Determine whether the point (3,8) is a solution to each system of equations. 2x + y = 14 x + y = 11 2(3) + 8 = 14 (YES) 3 + 8 = 11 (YES) YES! Both equations are true y = -x – 5 y = x + 5 8 = -3 – 5 (NO) 8 = 3 + 5 (YES) NO! Only 1 equation is true 4x – y = - 4 3x – 2y = 7 4(3) - 8 = - 4 (NO) 3(3) -2(8) = 7 (NO) NO! Neither equations are true

58. Practice Makes Perfect! On the whiteboard, solve the system graphically (use x & y intercepts to graph) and then verify the solution. Odd desk – do #1Even desk – do #2 1.2x + y = 82. -3x + 2y = 6 -x + y = 2 3x + y = 12 Bonus! 7x + 2y = 5 2x + 3y = 16 9/6/201557

59. Think-Pair-Share A Better Deal When the date for the work project was set, it turned out that only 13 science club members could participate. The club president talked again with the PTA president and got a new pay deal - $20 per outdoor worker and $15 per indoor worker. Still, $240 is needed for the club… 9/6/20159/6/20159/6/20159/6/20159/6/20159/6/20159/6/20159/6/2015 58 Smyrna High School Geometry CP Smyrna High School Geometry CP

60. REMIND YOUR GROUPIES Smyrna High School Geometry CP 59 Right: How do I graph an equation if it is not in y = form? Left: How do I know if my answer is correct? 9/6/2015 Middle: What is a solution to a system of equations?

61. Summarize 3-2-1 9/6/2015Smyrna High School Geometry CP 60

62. Summarize! Journal Time How can you solve a system of equations graphically if it is not in slope- intercept form? What are the steps? 9/6/201561Smyrna High School Geometry CP

63. Common Core Standards A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. F.BF.1 Write a function that describes a relationship between two quantities. ★ a. Determine an explicit expression, a recursive process, or steps for calculation from a context. 9/6/2015Smyrna High School Geometry CP 62