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

2. Linear Graphs and EquationsUnderstanding Gradient

3. Sketching linear graphs

4. Identifying graphsWhat makes a linear equation LINEAR?An equation in one or more variables, each with an exponent of ONLY 1, where these variables are only added or subtracted. For ex : 2x + 7 = 15

5. 4y + 5 = y + 15

6. 3y = -4x + 12, Tentukannilai x jika y = 0 ! So with that definition Which of these equations are linear?x+y = 52x+ 3y = 47x-3y = 14y = 2x-2y=4 x2 + y = 5x = 5xy = 5x2 +y2 = 9y = x2y3

7. So with that definition Which of these equations are linear?LinearNot Linearx+y = 52x+ 3y = 47x-3y = 14y = 2x-2y=4 x2 + y = 5x = 5xy = 5x2 +y2 = 9y = x2y3

9. yxLinearNot LinearWhat is a Linear Equation?A linear equation is an equation whose graph is a LINE.

10. yxWhat is a Linear Equation?The equations we will be graphing have two variables, x and y.4For example,2A solution to the equation is any ordered pair (x , y) that makes the equation true. -33-1-216The 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. yxThe x - values are picked by YOU!Graphing a Linear EquationHow do we graph linear equations?Let’s try this one: y = 3x – 2Make a Table of values–8y = 3(–2) – 2 = –8Complete the table by inputting the x - values and calculating the corresponding y - values.–5y = 3(–1) – 2 = –5–2y = 3(0) – 2 = –21y = 3(1) – 2 = 14y = 3(2) – 2 = 4

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

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

14. Sketching Linear GraphsWhat 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 Graphs2-4We 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. yxTake a moment and complete the chart…Click the screen when finishedGraphing a Linear EquationHow about another one!Let’s try 4x – 3y = 12To makes things easier:Make a Table of Values-134x – 3y = 120-40-4

18. yxGraphing Horizontal & Vertical LinesWhen 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 = 3y = –2Since there are no y – values in this equation, x is always 3 and y can be any other real number. Graph y = –2Since 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. SlopeParallel linesTheir slopes will be EQUAL.Perpendicular linesTheir slopes will be the negative reciprocal of each other.

22. Increase in yGradient =Increase in xGradient / SlopeGradient tells us how steep something is.

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

24. GradientIncrease in yGradient =Increase in xWhat is the gradient of the line?Gradient = 5/2 or 2.5

25. GradientIncrease in yGradient =Increase in xWhat is the gradient of this line?This time there is a decrease in yGradient = -2/4 or -0.5

27. Exercise 2

29. yx4321-10123-1-2

30. Linear EquationsStandard Form Ax + By = C Ex ; 2x + y = 3Slope-intercept form y = mx + bm = slope/gradientb = y-intercept

32. Contoh5x – 2y = 6 (Standard Form)– 2y = 6 – 5xy = 6 – 5x – 2 y = 6 – 5x – 2 – 2y = - 3 + 5/2 x y = 5/2 x – 3 (Slope, y-Intercept)

33. Slope-intercept formy - interceptGradientEx ; y = -2x + 3 (Slope, y-Intercept)Linear Graphs Form y = mxdan y = mx + bPage 84 - 87

35. MenentukanBentukPersamaanjikadiketahuigrafikVideo 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 = -42. Find the y-intercept. y = mx + b 2 = -4 • 4 + b 2 = -16 + b18 = b

40. (4, 2) and (3, 6) m = -4 b = 183. Write equation in y = mx + b.y = -4x + 184. Convert to Ax + By = C.4x + y = 18

41. Linear EquationsBe able to form an equation given… - slope and y-interceptex. m = -3 and b = 5 - a point and the slopeex. ( -4, -1 ) and m = ¾ - two pointsex. ( 0, -4 ) and ( -5, -2 )

42. MenentukanBentukPersamaanjikadiketahuigrafikVideo Contoh

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 interceptFalling-3/4 -66/(-3/4) = - 8 -3/4 4/33x + 4y = -6The Equation FormDirectionSlopey-interceptx-interceptParallel SlopePerpendicular SlopeStandard Form

52. Given our 4 example equations identify all of the following…The Equation FormDirectionSlopey-interceptx-interceptParallel SlopePerpendicular SlopeFormy = ½ x + 5y = -3x – 73x – 2y = 94x + 2y = 16x – 6y + 1 = 0

53. y = ½ x + 5Slope interceptRising½ 5-5/(½) = -10½ -2- x +2y = 5The Equation FormDirectionSlopey-interceptx-interceptParallel SlopePerpendicular SlopeStandard Form

54. y = -3x – 7Slope interceptFalling-3 -7- -7/(-3) = -7/3-3 -73x + y = - 7The Equation FormDirectionSlopey-interceptx-interceptParallel SlopePerpendicular SlopeStandard Form

55. 3x – 2y = 9StandardRising3/2 -4.5 or 9/233/2 -2/3y =3/2x + 9/2The Equation FormDirectionSlopey-interceptx-interceptParallel SlopePerpendicular SlopeSlope,intercept Form

56. 4x + 2y = 16StandardFalling-2 84-2 1/2y = -2x + 8The Equation FormDirectionSlopey-interceptx-interceptParallel SlopePerpendicular SlopeSlope,Intercept Form

57. GeneralFalling½ 2-1½ -2y = ½ x + 2The Equation FormDirectionSlopey-interceptx-interceptParallel SlopePerpendicular SlopeSlope,intercept Formx – 2y +4= 0

58. Exercise 6

59. Drawing with slope

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

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

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

64. Soal 4Tentukanpersamaangaris 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 + bLangkah 4 BentukPersamaanGaris !

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