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# 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