Linear equations
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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.
Linear equations
- 2. Linear Graphs and EquationsUnderstanding Gradient
- 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?Lets try this one: y = 3x 2Make a Table of values8y = 3(2) 2 = 8Complete the table by inputting the x - values and calculating the corresponding y - values.5y = 3(1) 2 = 52y = 3(0) 2 = 21y = 3(1) 2 = 14y = 3(2) 2 = 4
- 12. yxGraphing a Linear EquationHow about another one!Lets try x 2y = 5.First Step:Write y as a function of xx 2y = 52y = 5 x
- 13. yxTake a moment and complete the chartClick the screen when finishedGraphing a Linear EquationHow about another one!Lets try x 2y = 5.Second Step:Make a Table of Values32
- 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 chartClick the screen when finishedGraphing a Linear EquationHow about another one!Lets 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 doesnt 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
- 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 = 他 - two pointsex. ( 0, -4 ) and ( -5, -2 )
- 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 followingThe Equation FormDirectionSlopey-interceptx-interceptParallel SlopePerpendicular SlopeFormy = 遜 x + 5y = -3x 73x 2y = 94x + 2y = 16x 6y + 1 = 0
- 53. y = 遜 x + 5Slope interceptRising遜 5-5/(遜) = -10遜 -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. GeneralFalling遜 2-1遜 -2y = 遜 x + 2The Equation FormDirectionSlopey-interceptx-interceptParallel SlopePerpendicular SlopeSlope,intercept Formx 2y +4= 0
- 58. Exercise 6
- 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 adalahJawab :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 !
- 66. SOAL 6Tentukanpersamaangaris yang sejajardengangaris y = x + 8 danmelaluititik (-2, 3)Jawab : Langkah 1 CariGradien (m) daripersamaangarisLangkah 2 Ingat ! Sejajar m-nyatetapLangkah 3 gunakan y = mx + b