1. Linear Equations

3. Sketching linear graphs

5. 4y + 5 = y + 15

7. So with that definition Which of these equations are linear? Linear Not Linear x+y = 5 2x+ 3y = 4 7x-3y = 14 y = 2x-2 y=4 x2 + y = 5 x = 5 xy = 5 x2 +y2 = 9 y = x2 y 3

9. y x Linear Not Linear What is a Linear Equation? A linear equation is an equation whose graph is a LINE.

10. y x What is a Linear Equation? The equations we will be graphing have two variables, x and y. 4 For example, 2 A solution to the equation is any ordered pair (x , y) that makes the equation true. -3 3 -1 -2 1 6 The ordered pair (3 , 2) is a solution since, If we were to plot all these ordered pairs on a graph, we would be graphing a line.

11. y x The x - values are picked by YOU! Graphing a Linear Equation How do we graph linear equations? Let’s try this one: y = 3x – 2 Make a Table of values –8 y = 3(–2) – 2 = –8 Complete the table by inputting the x - values and calculating the corresponding y - values. –5 y = 3(–1) – 2 = –5 –2 y = 3(0) – 2 = –2 1 y = 3(1) – 2 = 1 4 y = 3(2) – 2 = 4

12. y x Graphing a Linear Equation How about another one! Let’s try x – 2y = 5. First Step: Write y as a function of x x – 2y = 5 –2y = 5 – x

13. y x Take a moment and complete the chart… Click the screen when finished Graphing a Linear Equation How about another one! Let’s try x – 2y = 5. Second Step: Make a Table of Values –3 –2

14. Sketching Linear Graphs What is y when x is 0? What is x when y is 0? We can now use this to get two sets of coordinates.

15. Sketching Linear Graphs 2 -4 We know that our line must go through the points (0,-4) and (2,0) To draw a sketch of this graph, we just need to label the important points.

17. y x Take a moment and complete the chart… Click the screen when finished Graphing a Linear Equation How about another one! Let’s try 4x – 3y = 12 To makes things easier: Make a Table of Values -1 3 4x – 3y = 12 0 -4 0 -4

18. y x Graphing Horizontal & Vertical Lines When you are asked to graph a line, and there is only ONE variable in the equation, the line will either be vertical or horizontal. For example … Graph x = 3 y = –2 Since there are no y – values in this equation, x is always 3 and y can be any other real number. Graph y = –2 Since there are no x – values in this equation, y is always – 2 and x can be any other real number. x = 3

20. Exercise 1

21. Slope Parallel lines Their slopes will be EQUAL. Perpendicular lines Their slopes will be the negative reciprocal of each other.

23. For ex; Page 75This child doesn’t have a clue about gradient.

24. Gradient Increase in y Gradient = Increase in x What is the gradient of the line? Gradient = 5/2 or 2.5

25. Gradient Increase in y Gradient = Increase in x What is the gradient of this line? This time there is a decrease in y Gradient = -2/4 or -0.5

27. Exercise 2

29. y x 4 3 2 1 -1 0 1 2 3 -1 -2

30. Linear Equations Standard Form Ax + By = C Ex ; 2x + y = 3 Slope-intercept form y = mx + b m = slope/gradient b = y-intercept

32. Contoh 5x – 2y = 6 (Standard Form) – 2y = 6 – 5x y = 6 – 5x – 2 y = 6 – 5x – 2 – 2 y = - 3 + 5/2 x y = 5/2 x – 3 (Slope, y-Intercept)

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36. Exercise 4

37. A Point and The Slope

38. Always find slope-intercept form first! Find the equation for the line containing the points (4, 2) and (3, 6). Find the slope using the formula. m = 2 – 6 4 – 3 m = -4

39. (4, 2) and (3, 6)m = -4 2. Find the y-intercept. y = mx + b 2 = -4 • 4 + b 2 = -16 + b 18 = b

40. (4, 2) and (3, 6) m = -4 b = 18 3. Write equation in y = mx + b. y = -4x + 18 4. Convert to Ax + By = C. 4x + y = 18

41. Linear Equations Be able to form an equation given… - slope and y-intercept ex. m = -3 and b = 5 - a point and the slope ex. ( -4, -1 ) and m = ¾ - two points ex. ( 0, -4 ) and ( -5, -2 )

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44. Latihan 1

45. Latihan 2

46. Graph Form

47. Table form

48. Exercise 5

49. Relationship between Slope and Linear equations Pertemuan ke-6

51. y = -3/4 x – 6 Slope intercept Falling -3/4 -6 6/(-3/4) = - 8 -3/4 4/3 3x + 4y = -6 The Equation Form Direction Slope y-intercept x-intercept Parallel Slope Perpendicular Slope Standard Form

52. Given our 4 example equations identify all of the following… The Equation Form Direction Slope y-intercept x-intercept Parallel Slope Perpendicular Slope Form y = ½ x + 5 y = -3x – 7 3x – 2y = 9 4x + 2y = 16 x – 6y + 1 = 0

53. y = ½ x + 5 Slope intercept Rising ½ 5 -5/(½) = -10 ½ -2 - x +2y = 5 The Equation Form Direction Slope y-intercept x-intercept Parallel Slope Perpendicular Slope Standard Form

54. y = -3x – 7 Slope intercept Falling -3 -7 - -7/(-3) = -7/3 -3 -7 3x + y = - 7 The Equation Form Direction Slope y-intercept x-intercept Parallel Slope Perpendicular Slope Standard Form

55. 3x – 2y = 9 Standard Rising 3/2 -4.5 or 9/2 3 3/2 -2/3 y =3/2x + 9/2 The Equation Form Direction Slope y-intercept x-intercept Parallel Slope Perpendicular Slope Slope,intercept Form

56. 4x + 2y = 16 Standard Falling -2 8 4 -2 1/2 y = -2x + 8 The Equation Form Direction Slope y-intercept x-intercept Parallel Slope Perpendicular Slope Slope,Intercept Form

57. General Falling ½ 2 -1 ½ -2 y = ½ x + 2 The Equation Form Direction Slope y-intercept x-intercept Parallel Slope Perpendicular Slope Slope,intercept Form x – 2y +4= 0

58. Exercise 6

59. Drawing with slope

61. SOAL 1 Tentukanpersamaangaris yang tegaklurusdengangaris4x – 3y– 6 = 0 danmelaluititik (2, -3) Jawab : Langkah 1 CariGradien (m) denganmembuatpersamaangarisbentukgradien Langkah 2 Ingat !!! TegakLurus (Rubahgradiennya !!!) Langkah 3 gunakan y = mx + b

62. SOAL 2 Hubungangaris3x + 4y – 6 = 0 dengangaris-6y = -8x +10 adalah… Jawab : Langkah 1 Carimdarikeduapersamaan Langkah 2 Sederhanakan, tentukansejajar/ berpotongantegaklurus !

63. Soal 3 Garis 2x +5y – 2 = 0 sejajardengangaris 3ax – 4y – 2 = 0, tentukannilaia! Jawab : Langkah 1 Carimdaripersamaangarisygsudahdiketahui Langkah 2 Ingat !!! m-nyaSejajar Langkah 3 padapersamaangaris 3ax – 4y – 2 = 0, dibuatbentukgradien Langkah 4 Caria dari L.2 & L.3 !

64. Soal 4 Tentukanpersamaangaris yang melaluititik (-2, -3 ) dantegaklurusdengangaris yang melaluititik( 2,3 ) dan (0, 1) Jawab ; Langkah 1 carimdarititik ( 2,3 ) dan(0, 1) Langkah 2 ingattegaklurus m-nyadirubah !!! Langkah 3 cari b dengan y = mx + b Langkah 4 BentukPersamaanGaris !

65. Soal 5 Tentukanpersamaangaris yang melaluititik (-2, 1 ) dansejajardengangaris yang melaluititik ( 4,3 ) dan (-2,-5) Jawab ; Langkah 1 carimdarititik ( 2,3 ) dan (0, 1) Langkah 2 ingatSejajarm-nyaTetap Langkah 3 cari b dengan y = mx + b Langkah 4 BentukPersamaanGaris !

66. SOAL 6 Tentukanpersamaangaris yang sejajardengangaris y = x + 8 dan melaluititik (-2, 3) Jawab : Langkah 1 CariGradien (m) daripersamaangaris Langkah 2 Ingat ! Sejajar m-nyatetap Langkah 3 gunakan y = mx + b