The document discusses linear equations and graphs. It defines a linear equation as one where the variables each have an exponent of 1 and are only added or subtracted. It then identifies which of several example equations are linear based on this definition. The document explains that the graph of a linear equation is a straight line. It shows how to graph linear equations by making tables of values and plotting points. It also discusses how to graph vertical and horizontal lines when there is only one variable. Finally, it covers finding the equation of a line given its slope and y-intercept, or two points on the line.

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)

35. MenentukanBentukPersamaan jikadiketahuigrafik Video Contoh

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