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Linear Equations<br />
Linear Graphs and Equations<br /><ul><li>Understanding Gradient
Sketching linear graphs
Identifying graphs</li></li></ul><li>What makes a linear equation LINEAR?<br />An equation in one or more variables, each ...
4y + 5 = y + 15
3y = -4x + 12, Tentukannilai x jika y = 0 ! </li></li></ul><li>So with that definition Which of these equations are linear...
So with that definition Which of these equations are linear?<br />Linear<br />Not Linear<br />x+y = 5<br />2x+ 3y = 4<br /...
y<br />x<br />Linear<br />Not Linear<br />What is a Linear Equation?<br />A linear equation is an equation whose graph is ...
y<br />x<br />What is a Linear Equation?<br />The equations we will be graphing have two variables, x and y.<br />4<br />F...
y<br />x<br />The x - values are picked by YOU!<br />Graphing a Linear Equation<br />How do we graph linear equations?<br ...
y<br />x<br />Graphing a Linear Equation<br />How about another one!<br />Let’s try x – 2y = 5.<br />First Step:<br />Writ...
y<br />x<br />Take a moment and complete the chart…<br />Click the screen when finished<br />Graphing a Linear Equation<br...
Sketching Linear Graphs<br />What is y when x is 0?<br />What is x when y is 0?<br />We can now use this to get two sets o...
Sketching Linear Graphs<br />2<br />-4<br />We know that our line must go through the points (0,-4) and (2,0)<br />To draw...
y<br />x<br />Take a moment and complete the chart…<br />Click the screen when finished<br />Graphing a Linear Equation<br...
y<br />x<br />Graphing Horizontal & Vertical Lines<br />When you are asked to graph a line, and there is only ONE variable...
Exercise 1<br />
Slope<br />Parallel lines<br />Their slopes will be EQUAL.<br />Perpendicular lines<br />Their slopes will be the   negati...
Increase in y<br />Gradient =<br />Increase in x<br />Gradient / Slope<br /><ul><li>Gradient tells us how steep something is.
For ex; Page 75</li></ul>This child doesn’t have a clue about gradient.<br />
Gradient<br />Increase in y<br />Gradient =<br />Increase in x<br />What is the gradient of the line?<br />Gradient = 5/2 ...
Gradient<br />Increase in y<br />Gradient =<br />Increase in x<br />What is the gradient of this line?<br />This time ther...
Exercise 2<br />
y<br />x<br />4<br />3<br />2<br />1<br />-1<br />0<br />1<br />2<br />3<br />-1<br />-2<br />
Linear Equations<br />Standard Form	Ax + By = C<br />				Ex ;  2x + y = 3<br />Slope-intercept form	y = mx + b<br />m = sl...
Contoh<br />5x – 2y = 6 	(Standard Form)<br />– 2y	= 6 – 5x<br />y		= 6 – 5x<br />		    – 2	<br />y		=  6   –   5x<br />		...
Slope-intercept form<br />y - intercept<br />Gradient<br /><ul><li>Ex ;     y = -2x + 3 (Slope, y-Intercept)</li></li></ul...
MenentukanBentukPersamaan<br />jikadiketahuigrafik<br />Video Contoh<br />
Exercise 4<br />
A Point and The Slope<br />
Always find slope-intercept form first!<br />Find the equation for the line containing the points (4, 2) and (3, 6).<br />...
(4, 2) and (3, 6)m = -4<br />2. Find the y-intercept.<br />			y = mx + b<br />			2 = -4 • 4 + b<br />			2 = -16 + b<br />1...
(4, 2) and (3, 6) m = -4   b = 18<br />3. Write equation in y = mx + b.<br />y = -4x + 18<br />4. Convert to Ax + By = C.<...
Linear Equations<br />Be able to form an equation given…<br />		- slope and y-intercept<br />ex. m = -3 and b = 5<br />		-...
MenentukanBentukPersamaan<br />jikadiketahuigrafik<br />Video Contoh<br />
Latihan 1<br />
Latihan 2<br />
Graph Form<br />
Table form<br />
Exercise 5<br />
Relationship between Slope and Linear equations <br />Pertemuan ke-6<br />
y = -3/4 x – 6 <br />Slope intercept<br />Falling<br />-3/4 <br />-6<br />6/(-3/4) = - 8 <br />-3/4 <br />4/3<br />3x + 4y...
Given our 4 example equations identify all of the following…<br />The Equation Form<br />Direction<br />Slope<br />y-inter...
y = ½ x + 5<br />Slope intercept<br />Rising<br />½ <br />5<br />-5/(½) = -10<br />½ <br />-2<br />- x +2y = 5<br />The Eq...
y  = -3x – 7<br />Slope intercept<br />Falling<br />-3 <br />-7<br />- -7/(-3) = -7/3<br />-3 <br />-7<br />3x + y = - 7<b...
3x – 2y = 9<br />Standard<br />Rising<br />3/2 <br />-4.5 or 9/2<br />3<br />3/2 <br />-2/3<br />y =3/2x + 9/2<br />The Eq...
4x + 2y = 16<br />Standard<br />Falling<br />-2 <br />8<br />4<br />-2 <br />1/2<br />y = -2x + 8<br />The Equation Form<b...
General<br />Falling<br />½  <br />2<br />-1<br />½  <br />-2<br />y = ½ x + 2<br />The Equation Form<br />Direction<br />...
Exercise 6<br />
Drawing with slope<br />
SOAL 1<br />Tentukanpersamaangaris yang  tegaklurusdengangaris4x – 3y– 6 = 0 danmelaluititik (2, -3)<br />Jawab : <br />La...
SOAL 2<br />Hubungangaris3x + 4y – 6 = 0 dengangaris-6y = -8x +10 adalah…<br />Jawab :<br />Langkah 1		Carimdarikeduapersa...
Soal 3<br />Garis 2x +5y – 2 = 0 sejajardengangaris 3ax – 4y – 2 = 0, tentukannilaia!<br />Jawab :<br />Langkah 1		Carimda...
Soal 4<br />Tentukanpersamaangaris yang melaluititik (-2, -3 ) dantegaklurusdengangaris yang melaluititik( 2,3 ) dan (0, 1...
Soal 5<br />Tentukanpersamaangaris yang melaluititik (-2, 1 ) dansejajardengangaris yang melaluititik ( 4,3 ) dan (-2,-5) ...
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Linear equations

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Linear equations

  1. 1. Linear Equations<br />
  2. 2. Linear Graphs and Equations<br /><ul><li>Understanding Gradient
  3. 3. Sketching linear graphs
  4. 4. Identifying graphs</li></li></ul><li>What makes a linear equation LINEAR?<br />An equation in one or more variables, each with an exponent of ONLY 1, where these variables are only added or subtracted. <br /><ul><li>For ex : 2x + 7 = 15
  5. 5. 4y + 5 = y + 15
  6. 6. 3y = -4x + 12, Tentukannilai x jika y = 0 ! </li></li></ul><li>So with that definition Which of these equations are linear?<br />x+y = 5<br />2x+ 3y = 4<br />7x-3y = 14<br />y = 2x-2<br />y=4 <br />x2 + y = 5<br />x = 5<br />xy = 5<br />x2 +y2 = 9<br />y = x2<br />y<br />3<br />
  7. 7. So with that definition Which of these equations are linear?<br />Linear<br />Not Linear<br />x+y = 5<br />2x+ 3y = 4<br />7x-3y = 14<br />y = 2x-2<br />y=4 <br />x2 + y = 5<br />x = 5<br />xy = 5<br />x2 +y2 = 9<br />y = x2<br />y<br />3<br />
  8. 8.
  9. 9. y<br />x<br />Linear<br />Not Linear<br />What is a Linear Equation?<br />A linear equation is an equation whose graph is a LINE.<br />
  10. 10. y<br />x<br />What is a Linear Equation?<br />The equations we will be graphing have two variables, x and y.<br />4<br />For example,<br />2<br />A solution to the equation is any ordered pair (x , y) that makes the equation true. <br />-3<br />3<br />-1<br />-2<br />1<br />6<br />The ordered pair (3 , 2) is a solution since,<br />If we were to plot all these ordered pairs on a graph, we would be graphing a line.<br />
  11. 11. y<br />x<br />The x - values are picked by YOU!<br />Graphing a Linear Equation<br />How do we graph linear equations?<br />Let’s try this one: y = 3x – 2<br />Make a Table of values<br />–8<br />y = 3(–2) – 2 = –8<br />Complete the table by inputting the x - values and calculating the corresponding y - values.<br />–5<br />y = 3(–1) – 2 = –5<br />–2<br />y = 3(0) – 2 = –2<br />1<br />y = 3(1) – 2 = 1<br />4<br />y = 3(2) – 2 = 4<br />
  12. 12. y<br />x<br />Graphing a Linear Equation<br />How about another one!<br />Let’s try x – 2y = 5.<br />First Step:<br />Write y as a function of x<br />x – 2y = 5<br />–2y = 5 – x<br />
  13. 13. y<br />x<br />Take a moment and complete the chart…<br />Click the screen when finished<br />Graphing a Linear Equation<br />How about another one!<br />Let’s try x – 2y = 5.<br />Second Step:<br />Make a Table of Values<br />–3<br />–2<br />
  14. 14. Sketching Linear Graphs<br />What is y when x is 0?<br />What is x when y is 0?<br />We can now use this to get two sets of coordinates.<br />
  15. 15. Sketching Linear Graphs<br />2<br />-4<br />We know that our line must go through the points (0,-4) and (2,0)<br />To draw a sketch of this graph, we just need to label the important points.<br />
  16. 16.
  17. 17. y<br />x<br />Take a moment and complete the chart…<br />Click the screen when finished<br />Graphing a Linear Equation<br />How about another one!<br />Let’s try 4x – 3y = 12<br />To makes things easier:<br />Make a Table of Values<br />-1<br />3<br />4x – 3y = 12<br />0<br />-4<br />0<br />-4<br />
  18. 18. y<br />x<br />Graphing Horizontal & Vertical Lines<br />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 …<br />Graph x = 3<br />y = –2<br />Since there are no y – values in this equation, x is always 3 and y can be any other real number. <br />Graph y = –2<br />Since there are no x – values in this equation, y is always – 2 and x can be any other real number. <br />x = 3<br />
  19. 19.
  20. 20. Exercise 1<br />
  21. 21. Slope<br />Parallel lines<br />Their slopes will be EQUAL.<br />Perpendicular lines<br />Their slopes will be the negative reciprocal of each other.<br />
  22. 22. Increase in y<br />Gradient =<br />Increase in x<br />Gradient / Slope<br /><ul><li>Gradient tells us how steep something is.
  23. 23. For ex; Page 75</li></ul>This child doesn’t have a clue about gradient.<br />
  24. 24. Gradient<br />Increase in y<br />Gradient =<br />Increase in x<br />What is the gradient of the line?<br />Gradient = 5/2 or 2.5<br />
  25. 25. Gradient<br />Increase in y<br />Gradient =<br />Increase in x<br />What is the gradient of this line?<br />This time there is a decrease in y<br />Gradient = -2/4 or -0.5<br />
  26. 26.
  27. 27. Exercise 2<br />
  28. 28.
  29. 29. y<br />x<br />4<br />3<br />2<br />1<br />-1<br />0<br />1<br />2<br />3<br />-1<br />-2<br />
  30. 30. Linear Equations<br />Standard Form Ax + By = C<br /> Ex ; 2x + y = 3<br />Slope-intercept form y = mx + b<br />m = slope/gradient<br />b = y-intercept<br />
  31. 31.
  32. 32. Contoh<br />5x – 2y = 6 (Standard Form)<br />– 2y = 6 – 5x<br />y = 6 – 5x<br /> – 2 <br />y = 6 – 5x<br /> – 2 – 2<br />y = - 3 + 5/2 x y = 5/2 x – 3 (Slope, y-Intercept)<br />
  33. 33. Slope-intercept form<br />y - intercept<br />Gradient<br /><ul><li>Ex ; y = -2x + 3 (Slope, y-Intercept)</li></li></ul><li>Linear Graphs Form y = mxdan y = mx + b<br />Page 84 - 87<br />
  34. 34.
  35. 35. MenentukanBentukPersamaan<br />jikadiketahuigrafik<br />Video Contoh<br />
  36. 36. Exercise 4<br />
  37. 37. A Point and The Slope<br />
  38. 38. Always find slope-intercept form first!<br />Find the equation for the line containing the points (4, 2) and (3, 6).<br />Find the slope using the formula.<br /> m = 2 – 6 <br /> 4 – 3 <br />m = -4<br />
  39. 39. (4, 2) and (3, 6)m = -4<br />2. Find the y-intercept.<br /> y = mx + b<br /> 2 = -4 • 4 + b<br /> 2 = -16 + b<br />18 = b<br />
  40. 40. (4, 2) and (3, 6) m = -4 b = 18<br />3. Write equation in y = mx + b.<br />y = -4x + 18<br />4. Convert to Ax + By = C.<br />4x + y = 18<br />
  41. 41. Linear Equations<br />Be able to form an equation given…<br /> - slope and y-intercept<br />ex. m = -3 and b = 5<br /> - a point and the slope<br />ex. ( -4, -1 ) and m = ¾<br /> - two points<br />ex. ( 0, -4 ) and ( -5, -2 )<br />
  42. 42. MenentukanBentukPersamaan<br />jikadiketahuigrafik<br />Video Contoh<br />
  43. 43.
  44. 44. Latihan 1<br />
  45. 45. Latihan 2<br />
  46. 46. Graph Form<br />
  47. 47. Table form<br />
  48. 48. Exercise 5<br />
  49. 49. Relationship between Slope and Linear equations <br />Pertemuan ke-6<br />
  50. 50.
  51. 51. y = -3/4 x – 6 <br />Slope intercept<br />Falling<br />-3/4 <br />-6<br />6/(-3/4) = - 8 <br />-3/4 <br />4/3<br />3x + 4y = -6<br />The Equation Form<br />Direction<br />Slope<br />y-intercept<br />x-intercept<br />Parallel Slope<br />Perpendicular Slope<br />Standard Form<br />
  52. 52. Given our 4 example equations identify all of the following…<br />The Equation Form<br />Direction<br />Slope<br />y-intercept<br />x-intercept<br />Parallel Slope<br />Perpendicular Slope<br />Form<br />y = ½ x + 5<br />y = -3x – 7<br />3x – 2y = 9<br />4x + 2y = 16<br />x – 6y + 1 = 0<br />
  53. 53. y = ½ x + 5<br />Slope intercept<br />Rising<br />½ <br />5<br />-5/(½) = -10<br />½ <br />-2<br />- x +2y = 5<br />The Equation Form<br />Direction<br />Slope<br />y-intercept<br />x-intercept<br />Parallel Slope<br />Perpendicular Slope<br />Standard Form<br />
  54. 54. y = -3x – 7<br />Slope intercept<br />Falling<br />-3 <br />-7<br />- -7/(-3) = -7/3<br />-3 <br />-7<br />3x + y = - 7<br />The Equation Form<br />Direction<br />Slope<br />y-intercept<br />x-intercept<br />Parallel Slope<br />Perpendicular Slope<br />Standard Form<br />
  55. 55. 3x – 2y = 9<br />Standard<br />Rising<br />3/2 <br />-4.5 or 9/2<br />3<br />3/2 <br />-2/3<br />y =3/2x + 9/2<br />The Equation Form<br />Direction<br />Slope<br />y-intercept<br />x-intercept<br />Parallel Slope<br />Perpendicular Slope<br />Slope,intercept Form<br />
  56. 56. 4x + 2y = 16<br />Standard<br />Falling<br />-2 <br />8<br />4<br />-2 <br />1/2<br />y = -2x + 8<br />The Equation Form<br />Direction<br />Slope<br />y-intercept<br />x-intercept<br />Parallel Slope<br />Perpendicular Slope<br />Slope,Intercept Form<br />
  57. 57. General<br />Falling<br />½ <br />2<br />-1<br />½ <br />-2<br />y = ½ x + 2<br />The Equation Form<br />Direction<br />Slope<br />y-intercept<br />x-intercept<br />Parallel Slope<br />Perpendicular Slope<br />Slope,intercept Form<br />x – 2y +4= 0<br />
  58. 58. Exercise 6<br />
  59. 59. Drawing with slope<br />
  60. 60.
  61. 61. SOAL 1<br />Tentukanpersamaangaris yang tegaklurusdengangaris4x – 3y– 6 = 0 danmelaluititik (2, -3)<br />Jawab : <br />Langkah 1 CariGradien (m) denganmembuatpersamaangarisbentukgradien<br />Langkah 2 Ingat !!! TegakLurus (Rubahgradiennya !!!)<br />Langkah 3 gunakan y = mx + b<br />
  62. 62. SOAL 2<br />Hubungangaris3x + 4y – 6 = 0 dengangaris-6y = -8x +10 adalah…<br />Jawab :<br />Langkah 1 Carimdarikeduapersamaan<br />Langkah 2 Sederhanakan, tentukansejajar/ berpotongantegaklurus !<br />
  63. 63. Soal 3<br />Garis 2x +5y – 2 = 0 sejajardengangaris 3ax – 4y – 2 = 0, tentukannilaia!<br />Jawab :<br />Langkah 1 Carimdaripersamaangarisygsudahdiketahui<br />Langkah 2 Ingat !!! m-nyaSejajar<br />Langkah 3 padapersamaangaris 3ax – 4y – 2 = 0, dibuatbentukgradien<br />Langkah 4 Caria dari L.2 & L.3 !<br />
  64. 64. Soal 4<br />Tentukanpersamaangaris yang melaluititik (-2, -3 ) dantegaklurusdengangaris yang melaluititik( 2,3 ) dan (0, 1) <br />Jawab ; <br />Langkah 1 carimdarititik ( 2,3 ) dan(0, 1) <br />Langkah 2 ingattegaklurus m-nyadirubah !!!<br />Langkah 3 cari b dengan y = mx + b<br />Langkah 4 BentukPersamaanGaris !<br />
  65. 65. Soal 5<br />Tentukanpersamaangaris yang melaluititik (-2, 1 ) dansejajardengangaris yang melaluititik ( 4,3 ) dan (-2,-5) <br />Jawab ; <br />Langkah 1 carimdarititik ( 2,3 ) dan (0, 1) <br />Langkah 2 ingatSejajarm-nyaTetap<br />Langkah 3 cari b dengan y = mx + b<br />Langkah 4 BentukPersamaanGaris !<br />
  66. 66. SOAL 6<br />Tentukanpersamaangaris yang sejajardengangaris y = x + 8 dan<br />melaluititik (-2, 3)<br />Jawab : <br />Langkah 1 CariGradien (m) daripersamaangaris<br />Langkah 2 Ingat ! Sejajar<br /> m-nyatetap<br />Langkah 3 gunakan y = mx + b<br />

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