3 2 linear equations and lines

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3 2 linear equations and lines

  1. 1. Linear Equations and Lines Back to 123a-Home
  2. 2. We solved 1st degree (linear) equations such as 2x + 1 = 5, which has a single variable x, to obtain its solution x = 2. Linear Equations and Lines
  3. 3. We solved 1st degree (linear) equations such as 2x + 1 = 5, which has a single variable x, to obtain its solution x = 2. Linear Equations and Lines We view this solution as the address of a position on a line and label it to produce a "picture“ of the answer:
  4. 4. We solved 1st degree (linear) equations such as 2x + 1 = 5, which has a single variable x, to obtain its solution x = 2. Linear Equations and Lines We view this solution as the address of a position on a line and label it to produce a "picture“ of the answer: 0 2 x The picture of x = 2
  5. 5. We solved 1st degree (linear) equations such as 2x + 1 = 5, which has a single variable x, to obtain its solution x = 2. Linear Equations and Lines We view this solution as the address of a position on a line and label it to produce a "picture“ of the answer: 0 2 x If we have a two–variable 1st degree equation such as 2x + y = 5 then we are free to select x and y. The picture of x = 2
  6. 6. We solved 1st degree (linear) equations such as 2x + 1 = 5, which has a single variable x, to obtain its solution x = 2. Linear Equations and Lines We view this solution as the address of a position on a line and label it to produce a "picture“ of the answer: 0 2 x If we have a two–variable 1st degree equation such as 2x + y = 5 then we are free to select x and y. For instance x = 2 and y = 1 make the equation true. The picture of x = 2
  7. 7. We solved 1st degree (linear) equations such as 2x + 1 = 5, which has a single variable x, to obtain its solution x = 2. Linear Equations and Lines We view this solution as the address of a position on a line and label it to produce a "picture“ of the answer: 0 2 x If we have a two–variable 1st degree equation such as 2x + y = 5 then we are free to select x and y. For instance x = 2 and y = 1 make the equation true. By viewing (2, 1) as the coordinate of a position in the xy-coordinate system, we have a picture of this solution. The picture of x = 2
  8. 8. We solved 1st degree (linear) equations such as 2x + 1 = 5, which has a single variable x, to obtain its solution x = 2. Linear Equations and Lines We view this solution as the address of a position on a line and label it to produce a "picture“ of the answer: 0 2 x If we have a two–variable 1st degree equation such as 2x + y = 5 then we are free to select x and y. For instance x = 2 and y = 1 make the equation true. By viewing (2, 1) as the coordinate of a position in the xy-coordinate system, we have a picture of this solution. (2, 1) The picture of x = 2 The picture of (x = 2, y = 1)
  9. 9. We solved 1st degree (linear) equations such as 2x + 1 = 5, which has a single variable x, to obtain its solution x = 2. Linear Equations and Lines We view this solution as the address of a position on a line and label it to produce a "picture“ of the answer: 0 2 x If we have a two–variable 1st degree equation such as 2x + y = 5 then we are free to select x and y. For instance x = 2 and y = 1 make the equation true. By viewing (2, 1) as the coordinate of a position in the xy-coordinate system, we have a picture of this solution. (2, 1) The picture of x = 2 Having the liberty of choosing two numbers means there are many pairs of solutions, thus more solution-points can be plotted. These points form the graph of the equation. The picture of (x = 2, y = 1)
  10. 10. In the rectangular coordinate system, ordered pairs (x, y)’s correspond to locations of points. Linear Equations and Lines
  11. 11. In the rectangular coordinate system, ordered pairs (x, y)’s correspond to locations of points. Collections of points may be specified by the mathematics relations between the x-coordinate and the y coordinate. Linear Equations and Lines
  12. 12. In the rectangular coordinate system, ordered pairs (x, y)’s correspond to locations of points. Collections of points may be specified by the mathematics relations between the x-coordinate and the y coordinate. The plot of points that fit a given relation is called the graph of that relation. Linear Equations and Lines
  13. 13. In the rectangular coordinate system, ordered pairs (x, y)’s correspond to locations of points. Collections of points may be specified by the mathematics relations between the x-coordinate and the y coordinate. The plot of points that fit a given relation is called the graph of that relation. To make a graph of a given mathematics relation, make a table of points that fit the description and plot them. Linear Equations and Lines
  14. 14. Example A. Graph the points (x, y) where x = –4 In the rectangular coordinate system, ordered pairs (x, y)’s correspond to locations of points. Collections of points may be specified by the mathematics relations between the x-coordinate and the y coordinate. The plot of points that fit a given relation is called the graph of that relation. To make a graph of a given mathematics relation, make a table of points that fit the description and plot them. Linear Equations and Lines
  15. 15. Example A. Graph the points (x, y) where x = –4 (y can be anything). In the rectangular coordinate system, ordered pairs (x, y)’s correspond to locations of points. Collections of points may be specified by the mathematics relations between the x-coordinate and the y coordinate. The plot of points that fit a given relation is called the graph of that relation. To make a graph of a given mathematics relation, make a table of points that fit the description and plot them. Linear Equations and Lines
  16. 16. Example A. Graph the points (x, y) where x = –4 (y can be anything). Make a table of ordered pairs of points that fit the description x = –4. In the rectangular coordinate system, ordered pairs (x, y)’s correspond to locations of points. Collections of points may be specified by the mathematics relations between the x-coordinate and the y coordinate. The plot of points that fit a given relation is called the graph of that relation. To make a graph of a given mathematics relation, make a table of points that fit the description and plot them. Linear Equations and Lines
  17. 17. Linear Equations and Lines Example A. Graph the points (x, y) where x = –4 (y can be anything). x y –4 –4 –4 –4 Make a table of ordered pairs of points that fit the description x = –4. In the rectangular coordinate system, ordered pairs (x, y)’s correspond to locations of points. Collections of points may be specified by the mathematics relations between the x-coordinate and the y coordinate. The plot of points that fit a given relation is called the graph of that relation. To make a graph of a given mathematics relation, make a table of points that fit the description and plot them. Linear Equations and Lines
  18. 18. Example A. Graph the points (x, y) where x = –4 (y can be anything). x y –4 0 –4 –4 –4 Make a table of ordered pairs of points that fit the description x = –4. In the rectangular coordinate system, ordered pairs (x, y)’s correspond to locations of points. Collections of points may be specified by the mathematics relations between the x-coordinate and the y coordinate. The plot of points that fit a given relation is called the graph of that relation. To make a graph of a given mathematics relation, make a table of points that fit the description and plot them. Linear Equations and Lines
  19. 19. Example A. Graph the points (x, y) where x = –4 (y can be anything). x y –4 0 –4 2 –4 –4 Make a table of ordered pairs of points that fit the description x = –4. In the rectangular coordinate system, ordered pairs (x, y)’s correspond to locations of points. Collections of points may be specified by the mathematics relations between the x-coordinate and the y coordinate. The plot of points that fit a given relation is called the graph of that relation. To make a graph of a given mathematics relation, make a table of points that fit the description and plot them. Linear Equations and Lines
  20. 20. Example A. Graph the points (x, y) where x = –4 (y can be anything). x y –4 0 –4 2 –4 4 –4 6 Make a table of ordered pairs of points that fit the description x = –4. In the rectangular coordinate system, ordered pairs (x, y)’s correspond to locations of points. Collections of points may be specified by the mathematics relations between the x-coordinate and the y coordinate. The plot of points that fit a given relation is called the graph of that relation. To make a graph of a given mathematics relation, make a table of points that fit the description and plot them. Linear Equations and Lines
  21. 21. Example A. Graph the points (x, y) where x = –4 (y can be anything). x y –4 0 –4 2 –4 4 –4 6 Graph of x = –4 Make a table of ordered pairs of points that fit the description x = –4. In the rectangular coordinate system, ordered pairs (x, y)’s correspond to locations of points. Collections of points may be specified by the mathematics relations between the x-coordinate and the y coordinate. The plot of points that fit a given relation is called the graph of that relation. To make a graph of a given mathematics relation, make a table of points that fit the description and plot them. Linear Equations and Lines
  22. 22. Example B. Graph the points (x, y) where y = x. Linear Equations and Lines
  23. 23. Example B. Graph the points (x, y) where y = x. Make a table of points that fit the description y = x. Linear Equations and Lines
  24. 24. Example B. Graph the points (x, y) where y = x. Make a table of points that fit the description y = x. To find one such point, we set one of the coordinates to be a number, any number, than use the relation to find the other coordinate. Linear Equations and Lines
  25. 25. Example B. Graph the points (x, y) where y = x. Make a table of points that fit the description y = x. To find one such point, we set one of the coordinates to be a number, any number, than use the relation to find the other coordinate. Repeat this a few times. Linear Equations and Lines
  26. 26. x y -1 0 1 2 Example B. Graph the points (x, y) where y = x. Make a table of points that fit the description y = x. To find one such point, we set one of the coordinates to be a number, any number, than use the relation to find the other coordinate. Repeat this a few times. Linear Equations and Lines
  27. 27. x y -1 -1 0 1 2 Example B. Graph the points (x, y) where y = x. Make a table of points that fit the description y = x. To find one such point, we set one of the coordinates to be a number, any number, than use the relation to find the other coordinate. Repeat this a few times. Linear Equations and Lines
  28. 28. x y -1 -1 0 0 1 2 Example B. Graph the points (x, y) where y = x. Make a table of points that fit the description y = x. To find one such point, we set one of the coordinates to be a number, any number, than use the relation to find the other coordinate. Repeat this a few times. Linear Equations and Lines
  29. 29. x y -1 -1 0 0 1 1 2 2 Example B. Graph the points (x, y) where y = x. Make a table of points that fit the description y = x. To find one such point, we set one of the coordinates to be a number, any number, than use the relation to find the other coordinate. Repeat this a few times. Linear Equations and Lines
  30. 30. x y -1 -1 0 0 1 1 2 2 Example B. Graph the points (x, y) where y = x. Make a table of points that fit the description y = x. To find one such point, we set one of the coordinates to be a number, any number, than use the relation to find the other coordinate. Repeat this a few times. Graph the points (x, y) where y = x Linear Equations and Lines
  31. 31. x y -1 -1 0 0 1 1 2 2 Example B. Graph the points (x, y) where y = x. Make a table of points that fit the description y = x. To find one such point, we set one of the coordinates to be a number, any number, than use the relation to find the other coordinate. Repeat this a few times. Graph the points (x, y) where y = x Linear Equations and Lines
  32. 32. x y -1 -1 0 0 1 1 2 2 Example B. Graph the points (x, y) where y = x. Make a table of points that fit the description y = x. To find one such point, we set one of the coordinates to be a number, any number, than use the relation to find the other coordinate. Repeat this a few times. Graph the points (x, y) where y = x Linear Equations and Lines
  33. 33. x y -1 -1 0 0 1 1 2 2 Example B. Graph the points (x, y) where y = x. Make a table of points that fit the description y = x. To find one such point, we set one of the coordinates to be a number, any number, than use the relation to find the other coordinate. Repeat this a few times. Graph the points (x, y) where y = x
  34. 34. First degree equation in the variables x and y are equations that may be put into the form Ax + By = C where A, B, C are numbers. Linear Equations and Lines
  35. 35. First degree equation in the variables x and y are equations that may be put into the form Ax + By = C where A, B, C are numbers. First degree equations are the same as linear equations. Linear Equations and Lines
  36. 36. First degree equation in the variables x and y are equations that may be put into the form Ax + By = C where A, B, C are numbers. First degree equations are the same as linear equations. They are called linear because their graphs are straight lines. Linear Equations and Lines
  37. 37. First degree equation in the variables x and y are equations that may be put into the form Ax + By = C where A, B, C are numbers. First degree equations are the same as linear equations. They are called linear because their graphs are straight lines. To graph a linear equation, find a few ordered pairs that fit the equation. Linear Equations and Lines
  38. 38. First degree equation in the variables x and y are equations that may be put into the form Ax + By = C where A, B, C are numbers. First degree equations are the same as linear equations. They are called linear because their graphs are straight lines. To graph a linear equation, find a few ordered pairs that fit the equation. To find one such ordered pair, assign a value to x, Linear Equations and Lines
  39. 39. First degree equation in the variables x and y are equations that may be put into the form Ax + By = C where A, B, C are numbers. First degree equations are the same as linear equations. They are called linear because their graphs are straight lines. To graph a linear equation, find a few ordered pairs that fit the equation. To find one such ordered pair, assign a value to x, plug it into the equation and solve for the y Linear Equations and Lines
  40. 40. First degree equation in the variables x and y are equations that may be put into the form Ax + By = C where A, B, C are numbers. First degree equations are the same as linear equations. They are called linear because their graphs are straight lines. To graph a linear equation, find a few ordered pairs that fit the equation. To find one such ordered pair, assign a value to x, plug it into the equation and solve for the y (or assign a value to y and solve for the x). Linear Equations and Lines
  41. 41. First degree equation in the variables x and y are equations that may be put into the form Ax + By = C where A, B, C are numbers. First degree equations are the same as linear equations. They are called linear because their graphs are straight lines. To graph a linear equation, find a few ordered pairs that fit the equation. To find one such ordered pair, assign a value to x, plug it into the equation and solve for the y (or assign a value to y and solve for the x). For graphing lines, find at least two ordered pairs. Linear Equations and Lines
  42. 42. First degree equation in the variables x and y are equations that may be put into the form Ax + By = C where A, B, C are numbers. First degree equations are the same as linear equations. They are called linear because their graphs are straight lines. To graph a linear equation, find a few ordered pairs that fit the equation. To find one such ordered pair, assign a value to x, plug it into the equation and solve for the y (or assign a value to y and solve for the x). For graphing lines, find at least two ordered pairs. Example C. Graph the following linear equations. a. y = 2x – 5 Linear Equations and Lines
  43. 43. First degree equation in the variables x and y are equations that may be put into the form Ax + By = C where A, B, C are numbers. First degree equations are the same as linear equations. They are called linear because their graphs are straight lines. To graph a linear equation, find a few ordered pairs that fit the equation. To find one such ordered pair, assign a value to x, plug it into the equation and solve for the y (or assign a value to y and solve for the x). For graphing lines, find at least two ordered pairs. Example C. Graph the following linear equations. a. y = 2x – 5 Make a table by selecting a few numbers for x. Linear Equations and Lines
  44. 44. First degree equation in the variables x and y are equations that may be put into the form Ax + By = C where A, B, C are numbers. First degree equations are the same as linear equations. They are called linear because their graphs are straight lines. To graph a linear equation, find a few ordered pairs that fit the equation. To find one such ordered pair, assign a value to x, plug it into the equation and solve for the y (or assign a value to y and solve for the x). For graphing lines, find at least two ordered pairs. Example C. Graph the following linear equations. a. y = 2x – 5 Make a table by selecting a few numbers for x. For easy caluation we set x = -1, 0, 1, and 2. Linear Equations and Lines
  45. 45. First degree equation in the variables x and y are equations that may be put into the form Ax + By = C where A, B, C are numbers. First degree equations are the same as linear equations. They are called linear because their graphs are straight lines. To graph a linear equation, find a few ordered pairs that fit the equation. To find one such ordered pair, assign a value to x, plug it into the equation and solve for the y (or assign a value to y and solve for the x). For graphing lines, find at least two ordered pairs. Example C. Graph the following linear equations. a. y = 2x – 5 Make a table by selecting a few numbers for x. For easy caluation we set x = -1, 0, 1, and 2. Plug each of these value into x and find its corresponding y to form an ordered pair. Linear Equations and Lines
  46. 46. For y = 2x – 5: x y -1 0 1 2 Linear Equations and Lines
  47. 47. For y = 2x – 5: x y -1 0 1 2 If x = -1, then y = 2(-1) – 5 Linear Equations and Lines
  48. 48. For y = 2x – 5: x y -1 -7 0 1 2 If x = -1, then y = 2(-1) – 5 = -7 Linear Equations and Lines
  49. 49. For y = 2x – 5: x y -1 -7 0 -5 1 2 If x = -1, then y = 2(-1) – 5 = -7 If x = 0, then y = 2(0) – 5 Linear Equations and Lines
  50. 50. For y = 2x – 5: x y -1 -7 0 -5 1 2 If x = -1, then y = 2(-1) – 5 = -7 If x = 0, then y = 2(0) – 5 = -5 Linear Equations and Lines
  51. 51. For y = 2x – 5: x y -1 -7 0 -5 1 -3 2 -1 If x = -1, then y = 2(-1) – 5 = -7 If x = 0, then y = 2(0) – 5 = -5 If x = 1, then y = 2(1) – 5 = -3 If x = 2, then y = 2(2) – 5 = -1 Linear Equations and Lines
  52. 52. For y = 2x – 5: x y -1 -7 0 -5 1 -3 2 -1 If x = -1, then y = 2(-1) – 5 = -7 If x = 0, then y = 2(0) – 5 = -5 If x = 1, then y = 2(1) – 5 = -3 If x = 2, then y = 2(2) – 5 = -1 Linear Equations and Lines
  53. 53. For y = 2x – 5: x y -1 -7 0 -5 1 -3 2 -1 If x = -1, then y = 2(-1) – 5 = -7 If x = 0, then y = 2(0) – 5 = -5 If x = 1, then y = 2(1) – 5 = -3 If x = 2, then y = 2(2) – 5 = -1 Linear Equations and Lines
  54. 54. For y = 2x – 5: x y -1 -7 0 -5 1 -3 2 -1 If x = -1, then y = 2(-1) – 5 = -7 If x = 0, then y = 2(0) – 5 = -5 If x = 1, then y = 2(1) – 5 = -3 If x = 2, then y = 2(2) – 5 = -1 Linear Equations and Lines
  55. 55. For y = 2x – 5: x y -1 -7 0 -5 1 -3 2 -1 If x = -1, then y = 2(-1) – 5 = -7 If x = 0, then y = 2(0) – 5 = -5 If x = 1, then y = 2(1) – 5 = -3 If x = 2, then y = 2(2) – 5 = -1 Linear Equations and Lines
  56. 56. For y = 2x – 5: x y -1 -7 0 -5 1 -3 2 -1 If x = -1, then y = 2(-1) – 5 = -7 If x = 0, then y = 2(0) – 5 = -5 If x = 1, then y = 2(1) – 5 = -3 If x = 2, then y = 2(2) – 5 = -1 Linear Equations and Lines
  57. 57. b. -3y = 12 Linear Equations and Lines
  58. 58. b. -3y = 12 Simplify as y = -4 Make a table by selecting a few numbers for x. Linear Equations and Lines
  59. 59. b. -3y = 12 Simplify as y = -4 Make a table by selecting a few numbers for x. x y -3 0 3 6 Linear Equations and Lines
  60. 60. b. -3y = 12 Simplify as y = -4 Make a table by selecting a few numbers for x. However, y = -4 is always. x y -3 -4 0 -4 3 -4 6 -4 Linear Equations and Lines
  61. 61. b. -3y = 12 Simplify as y = -4 Make a table by selecting a few numbers for x. However, y = -4 is always. x y -3 -4 0 -4 3 -4 6 -4 Linear Equations and Lines
  62. 62. b. -3y = 12 Simplify as y = -4 Make a table by selecting a few numbers for x. However, y = -4 is always. x y -3 -4 0 -4 3 -4 6 -4 Linear Equations and Lines
  63. 63. b. -3y = 12 Simplify as y = -4 Make a table by selecting a few numbers for x. However, y = -4 is always. x y -3 -4 0 -4 3 -4 6 -4 Linear Equations and Lines
  64. 64. b. -3y = 12 Simplify as y = -4 c. 2x = 12 Make a table by selecting a few numbers for x. However, y = -4 is always. x y -3 -4 0 -4 3 -4 6 -4 Linear Equations and Lines
  65. 65. b. -3y = 12 Simplify as y = -4 c. 2x = 12 Make a table by selecting a few numbers for x. However, y = -4 is always. x y -3 -4 0 -4 3 -4 6 -4 Simplify as x = 6. Linear Equations and Lines
  66. 66. b. -3y = 12 Simplify as y = -4 c. 2x = 12 Make a table by selecting a few numbers for x. However, y = -4 is always. x y -3 -4 0 -4 3 -4 6 -4 Simplify as x = 6 Make a table. However the only selction for x is x = 6 Linear Equations and Lines
  67. 67. b. -3y = 12 Simplify as y = -4 c. 2x = 12 Make a table by selecting a few numbers for x. However, y = -4 is always. x y -3 -4 0 -4 3 -4 6 -4 Simplify as x = 6 Make a table. However the only selction for x is x = 6 x y 6 6 6 6 Linear Equations and Lines
  68. 68. b. -3y = 12 Simplify as y = -4 c. 2x = 12 Make a table by selecting a few numbers for x. However, y = -4 is always. x y -3 -4 0 -4 3 -4 6 -4 Simplify as x = 6 Make a table. However the only selction for x is x = 6 and y could be any number. x y 6 0 6 2 6 4 6 6 Linear Equations and Lines
  69. 69. b. -3y = 12 Simplify as y = -4 c. 2x = 12 Make a table by selecting a few numbers for x. However, y = -4 is always. x y -3 -4 0 -4 3 -4 6 -4 Simplify as x = 6 Make a table. However the only selction for x is x = 6 and y could be any number. x y 6 0 6 2 6 4 6 6 Linear Equations and Lines
  70. 70. b. -3y = 12 Simplify as y = -4 c. 2x = 12 Make a table by selecting a few numbers for x. However, y = -4 is always. x y -3 -4 0 -4 3 -4 6 -4 Simplify as x = 6 Make a table. However the only selction for x is x = 6 and y could be any number. x y 6 0 6 2 6 4 6 6 Linear Equations and Lines
  71. 71. b. -3y = 12 Simplify as y = -4 c. 2x = 12 Make a table by selecting a few numbers for x. However, y = -4 is always. x y -3 -4 0 -4 3 -4 6 -4 Simplify as x = 6 Make a table. However the only selction for x is x = 6 and y could be any number. x y 6 0 6 2 6 4 6 6 Linear Equations and Lines
  72. 72. b. -3y = 12 Simplify as y = -4 c. 2x = 12 Make a table by selecting a few numbers for x. However, y = -4 is always. x y -3 -4 0 -4 3 -4 6 -4 Simplify as x = 6 Make a table. However the only selction for x is x = 6 and y could be any number. x y 6 0 6 2 6 4 6 6 Linear Equations and Lines
  73. 73. Summary of the graphs of linear equations: Linear Equations and Lines
  74. 74. a. y = 2x – 5 Summary of the graphs of linear equations: Linear Equations and Lines
  75. 75. a. y = 2x – 5 If both variables x and y are present in the equation, the graph is a tilted line. Summary of the graphs of linear equations: Linear Equations and Lines
  76. 76. a. y = 2x – 5 If both variables x and y are present in the equation, the graph is a tilted line. Summary of the graphs of linear equations: Linear Equations and Lines
  77. 77. a. y = 2x – 5 b. -3y = 12 If both variables x and y are present in the equation, the graph is a tilted line. Summary of the graphs of linear equations: Linear Equations and Lines
  78. 78. a. y = 2x – 5 b. -3y = 12 If both variables x and y are present in the equation, the graph is a tilted line. If the equation has only y (no x), the graph is a horizontal line. Summary of the graphs of linear equations: Linear Equations and Lines
  79. 79. a. y = 2x – 5 b. -3y = 12 If both variables x and y are present in the equation, the graph is a tilted line. If the equation has only y (no x), the graph is a horizontal line. Summary of the graphs of linear equations: Linear Equations and Lines
  80. 80. a. y = 2x – 5 b. -3y = 12 c. 2x = 12 If both variables x and y are present in the equation, the graph is a tilted line. If the equation has only y (no x), the graph is a horizontal line. Summary of the graphs of linear equations: Linear Equations and Lines
  81. 81. a. y = 2x – 5 b. -3y = 12 c. 2x = 12 If both variables x and y are present in the equation, the graph is a tilted line. If the equation has only y (no x), the graph is a horizontal line. Summary of the graphs of linear equations: If the equation has only x (no y), the graph is a vertical line. Linear Equations and Lines
  82. 82. a. y = 2x – 5 b. -3y = 12 c. 2x = 12 If both variables x and y are present in the equation, the graph is a tilted line. If the equation has only y (no x), the graph is a horizontal line. Summary of the graphs of linear equations: If the equation has only x (no y), the graph is a vertical line. Linear Equations and Lines
  83. 83. The x-Intercepts is where the line crosses the x-axis; Linear Equations and Lines
  84. 84. The x-Intercepts is where the line crosses the x-axis. We set y = 0 in the equation to find the x-intercept. Linear Equations and Lines
  85. 85. The x-Intercepts is where the line crosses the x-axis. We set y = 0 in the equation to find the x-intercept. The y-Intercepts is where the line crosses the y-axis; Linear Equations and Lines
  86. 86. The x-Intercepts is where the line crosses the x-axis. We set y = 0 in the equation to find the x-intercept. The y-Intercepts is where the line crosses the y-axis. We set x = 0 in the equation to find the y-intercept. Linear Equations and Lines
  87. 87. The x-Intercepts is where the line crosses the x-axis. We set y = 0 in the equation to find the x-intercept. The y-Intercepts is where the line crosses the y-axis. We set x = 0 in the equation to find the y-intercept. Since two points determine a line, an easy method to graph linear equations is the intercept method, Linear Equations and Lines
  88. 88. The x-Intercepts is where the line crosses the x-axis. We set y = 0 in the equation to find the x-intercept. The y-Intercepts is where the line crosses the y-axis. We set x = 0 in the equation to find the y-intercept. Since two points determine a line, an easy method to graph linear equations is the intercept method, i.e. plot the x-intercept and the y intercept and the graph is the line that passes through them. Linear Equations and Lines
  89. 89. The x-Intercepts is where the line crosses the x-axis. We set y = 0 in the equation to find the x-intercept. The y-Intercepts is where the line crosses the y-axis. We set x = 0 in the equation to find the y-intercept. Example C. Graph 2x – 3y = 12 by the intercept method. Since two points determine a line, an easy method to graph linear equations is the intercept method, i.e. plot the x-intercept and the y intercept and the graph is the line that passes through them. Linear Equations and Lines
  90. 90. x y 0 0 The x-Intercepts is where the line crosses the x-axis. We set y = 0 in the equation to find the x-intercept. The y-Intercepts is where the line crosses the y-axis. We set x = 0 in the equation to find the y-intercept. y-int x-int Example C. Graph 2x – 3y = 12 by the intercept method. Since two points determine a line, an easy method to graph linear equations is the intercept method, i.e. plot the x-intercept and the y intercept and the graph is the line that passes through them. Linear Equations and Lines
  91. 91. x y 0 0 The x-Intercepts is where the line crosses the x-axis. We set y = 0 in the equation to find the x-intercept. The y-Intercepts is where the line crosses the y-axis. We set x = 0 in the equation to find the y-intercept. y-int x-int Example C. Graph 2x – 3y = 12 by the intercept method. Since two points determine a line, an easy method to graph linear equations is the intercept method, i.e. plot the x-intercept and the y intercept and the graph is the line that passes through them. If x = 0, we get 2(0) – 3y = 12 Linear Equations and Lines
  92. 92. x y 0 -4 0 The x-Intercepts is where the line crosses the x-axis. We set y = 0 in the equation to find the x-intercept. The y-Intercepts is where the line crosses the y-axis. We set x = 0 in the equation to find the y-intercept. y-int x-int Example C. Graph 2x – 3y = 12 by the intercept method. Since two points determine a line, an easy method to graph linear equations is the intercept method, i.e. plot the x-intercept and the y intercept and the graph is the line that passes through them. If x = 0, we get 2(0) – 3y = 12 so y = -4 Linear Equations and Lines
  93. 93. x y 0 -4 0 The x-Intercepts is where the line crosses the x-axis. We set y = 0 in the equation to find the x-intercept. The y-Intercepts is where the line crosses the y-axis. We set x = 0 in the equation to find the y-intercept. y-int x-int Example C. Graph 2x – 3y = 12 by the intercept method. Since two points determine a line, an easy method to graph linear equations is the intercept method, i.e. plot the x-intercept and the y intercept and the graph is the line that passes through them. If x = 0, we get 2(0) – 3y = 12 so y = -4 If y = 0, we get 2x – 3(0) = 12 Linear Equations and Lines
  94. 94. x y 0 -4 6 0 The x-Intercepts is where the line crosses the x-axis. We set y = 0 in the equation to find the x-intercept. The y-Intercepts is where the line crosses the y-axis. We set x = 0 in the equation to find the y-intercept. y-int x-int Example C. Graph 2x – 3y = 12 by the intercept method. Since two points determine a line, an easy method to graph linear equations is the intercept method, i.e. plot the x-intercept and the y intercept and the graph is the line that passes through them. If x = 0, we get 2(0) – 3y = 12 so y = -4 If y = 0, we get 2x – 3(0) = 12 so x = 6 Linear Equations and Lines
  95. 95. x y 0 -4 6 0 The x-Intercepts is where the line crosses the x-axis. We set y = 0 in the equation to find the x-intercept. The y-Intercepts is where the line crosses the y-axis. We set x = 0 in the equation to find the y-intercept. y-int x-int Example C. Graph 2x – 3y = 12 by the intercept method. Since two points determine a line, an easy method to graph linear equations is the intercept method, i.e. plot the x-intercept and the y intercept and the graph is the line that passes through them. If x = 0, we get 2(0) – 3y = 12 so y = -4 If y = 0, we get 2x – 3(0) = 12 so x = 6 Linear Equations and Lines
  96. 96. x y 0 -4 6 0 The x-Intercepts is where the line crosses the x-axis. We set y = 0 in the equation to find the x-intercept. The y-Intercepts is where the line crosses the y-axis. We set x = 0 in the equation to find the y-intercept. y-int x-int Example C. Graph 2x – 3y = 12 by the intercept method. Since two points determine a line, an easy method to graph linear equations is the intercept method, i.e. plot the x-intercept and the y intercept and the graph is the line that passes through them. If x = 0, we get 2(0) – 3y = 12 so y = -4 If y = 0, we get 2x – 3(0) = 12 so x = 6 Linear Equations and Lines
  97. 97. x y 0 -4 6 0 The x-Intercepts is where the line crosses the x-axis. We set y = 0 in the equation to find the x-intercept. The y-Intercepts is where the line crosses the y-axis. We set x = 0 in the equation to find the y-intercept. y-int x-int Example C. Graph 2x – 3y = 12 by the intercept method. Since two points determine a line, an easy method to graph linear equations is the intercept method, i.e. plot the x-intercept and the y intercept and the graph is the line that passes through them. If x = 0, we get 2(0) – 3y = 12 so y = -4 If y = 0, we get 2x – 3(0) = 12 so x = 6 Linear Equations and Lines
  98. 98. Exercise. A. Solve the indicated variable for each equation with the given assigned value. 1. x + y = 3 and x = –1, find y. 2. x – y = 3 and y = –1, find x. 3. 2x = 6 and y = –1, find x. 4. –y = 3 and x = 2, find y. 5. 2y = 3 – x and x = –2 , find y. 6. y = –x + 4 and x = –4, find y. 7. 2x – 3y = 1 and y = 3, find x. 8. 2x = 6 – 2y and y = –2, find x. 9. 3y – 2 = 3x and x = 2, find y. 10. 2x + 3y = 3 and x = 0, find y. 11. 2x + 3y = 3 and y = 0, find x. 12. 3x – 4y = 12 and x = 0, find y. 13. 3x – 4y = 12 and y = 0, find x. 14. 6 = 3x – 4y and y = –3, find x. Linear Equations and Lines
  99. 99. B. a. Complete the tables for each equation with given values. b. Plot the points from the table. c. Graph the line. 15. x + y = 3 16. 2y = 6 x y -3 0 3 x y 1 0 –1 17. x = –6 x y 0 –1 – 2 18. y = x – 3 x y 2 1 0 19. 2x – y = 2 20. 3y = 6 + 2x x y 2 0 –1 x y 1 0 –1 21. y = –6 x y 0 –1 – 2 22. 3y + 4x =12 x y 0 0 1 Linear Equations and Lines
  100. 100. C. Make a table for each equation with at least 3 ordered pairs. (remember that you get to select one entry in each row as shown in the tables above) then graph the line. 23. x – y = 3 24. 2x = 6 25. –y – 7= 0 26. 0 = 8 – 2x 27. y = –x + 4 28. 2x – 3 = 6 29. 2x = 6 – 2y 30. 4y – 12 = 3x 31. 2x + 3y = 3 32. –6 = 3x – 2y 33. 35. For problems 29, 30, 31 and 32, use the intercept-tables as shown to graph the lines. x y 0 0 intercept-table 36. Why can’t we use the above intercept method to graph the lines for problems 25, 26 or 33? 37. By inspection identify which equations give horizontal lines, which give vertical lines and which give tilted lines. 3x = 4y 34. 5x + 2y = –10 Linear Equations and Lines

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