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# Chapter 1 linear equations and straight lines

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### Chapter 1 linear equations and straight lines

1. 1. Chapter 1 Linear Equations and Straight Lines Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 of 71
2. 2. Outline 1.1 Coordinate Systems and Graphs 1.2 Linear Inequalities 1.3 The Intersection Point of a Pair of Lines 1.4 The Slope of a Straight Line 1.5 The Method of Least Squares Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 of 71
3. 3. Section 1.1 Coordinate Systems and Graphs Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 of 71
4. 4. Outline 1. 2. 3. 4. 5. Coordinate Line Coordinate Plane Graph of an Equation Linear Equation Standard Form of Linear Equation 6. Graph of x = a 7. Intercepts 8. Graph of y = mx + b Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 of 71
5. 5. Coordinate Line Construct a Cartesian coordinate system on a line by choosing an arbitrary point, O (the origin), on the line and a unit of distance along the line. Then assign to each point on the line a number that reflects its directed distance from the origin. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 5 5 of 71
6. 6. Example Coordinate Line Graph the points -3/5, 1/2 and 15/8 on a coordinate line. 1/2 -3/5 -4 -3 -2 -1 Origin 0 15/8 1 2 3 4 Unit length Positive numbers Negative numbers Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 6 6 of 71
7. 7. Coordinate Plane Construct a Cartesian coordinate system on a plane by drawing two coordinate lines, called the coordinate axes, perpendicular at the origin. The horizontal line is called the x-axis, and the vertical line is the y-axis. y y-axis O x-axis Origin Copyright © 2014, 2010, 2007 Pearson Education, Inc. x Slide 7 7 of 71
8. 8. Coordinate Plane: Points Each point of the plane is identified by a pair of numbers (a,b). The first number tells the number of units from the point to the y-axis. The second tells the number of units from the point to the x-axis. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 8 8 of 71
9. 9. Example Coordinate Plane Plot the points: (2,1), (-1,3), (-2,-1) and (0,-3). (-1,3) -1 y 3 2 (2,1) 1 x -1 (-1,-2) -2 -3 (0,-3) Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 9 9 of 71
10. 10. Graph of an Equation The collection of points (x,y) that satisfies an equation is called the graph of that equation. Every point on the graph will satisfy the equation if the first coordinate is substituted for every occurrence of x and the second coordinate is substituted for every occurrence of y in the equation. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 10 10 of 71
11. 11. Example Graph of an Equation Sketch the graph of the equation y = 2x - 1. y (2,3) x y -2 2(-2) - 1 = -5 -1 2(0) - 1 = -1 1 2(1) - 1 = 1 2 2(2) - 1 = 3 x 2(-1) - 1 = -3 0 (1,1) (0,-1) (-1,-3) (-2,-5) Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 11 11 of 71
12. 12. Linear Equation An equation that can be put in the form cx + dy = e (c, d, e constants) is called a linear equation in x and y. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 12 12 of 71
13. 13. Standard Form of Linear Equation The standard form of a linear equation is y = mx + b (m, b constants) if y can be solved for, or x=a (a constant) if y does not appear in the equation. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 13 13 of 71
14. 14. Example Standard Form Find the standard form of 8x - 4y = 4 and 2x = 6. (a) 8x - 4y = 4 (b) 2x = 6 8x - 4y = 4 - 4y = - 8x + 4 y = 2x - 1 2x = 6 x=3 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 14 14 of 71
15. 15. Graph of x = a The equation x = a graphs into a vertical line a units from the y-axis. y y x=2 x = -3 x Copyright © 2014, 2010, 2007 Pearson Education, Inc. x Slide 15 15 of 71
16. 16. Intercepts x-intercept: a point on the graph that has a ycoordinate of 0. This corresponds to a point where the graph intersects the x-axis. y-intercept: the point on the graph that has a xcoordinate of 0. This corresponds to the point where the graph intersects the y-axis. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 16 16 of 71
17. 17. Graph of y = mx + b To graph the equation y = mx + b: 1. Plot the y-intercept (0,b). 2. Plot some other point. [The most convenient choice is often the x-intercept.] 3. Draw a line through the two points. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 17 17 of 71
18. 18. Example Graph of Linear Equation Use the intercepts to graph y = 2x - 1. x-intercept: Let y = 0 y 0 = 2x - 1 x = 1/2 (1/2,0) x y-intercept: Let x = 0 y = 2(0) - 1 = -1 (0,-1) y = 2x - 1 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 18 18 of 71
19. 19. Summary Section 1.1  Cartesian coordinate systems associate a number with each point of a line and associate a pair of numbers with each point of a plane.  The collection of points in the plane that satisfy the equation ax + by = c lies on a straight line. After this equation is put into one of the standard forms y = mx + b or x = a, the graph is easily drawn. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 19 19 of 71
20. 20. Section 1.2 Linear Inequalities Copyright © 2014, 2010, 2007 Pearson Education, Inc. 20 of 71
21. 21. Outline 1. Definitions of Inequality Signs 2. Inequality Property 1 3. Inequality Property 2 4. Standard Form of Inequality 5. Graph of x > a or x < a 6. Graph of y > mx + b or y < mx + b 7. Graphing System of Linear Inequalities Copyright © 2014, 2010, 2007 Pearson Education, Inc. 21 of 71
22. 22. Definitions of Inequality Signs  a < b means a lies to the left of b on the number line.  a < b means a = b or a < b.  a > b means a lies to the right of b on the number line.  a > b means a = b or a > b. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 22 22 of 71
23. 23. Inequality Signs Example -4 -3 -2 -1 0 1 2 3 4 Which of the following statements are true? True 1<4 True -1 > -4 True 2<3 True 0 > -2 True 3>3 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 23 23 of 71
24. 24. Inequality Property 1 Inequality Property 1 Suppose that a < b and that c is any number. Then a + c < b + c. In other words, the same number can be added or subtracted from both sides of the inequality. Note: Inequality Property 1 also holds if < is replaced by >, < or >. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 24 24 of 71
25. 25. Example Inequality Property 1 Solve the inequality x + 5 < 2. Subtract 5 from both sides to isolate the x on the left. x+5<2 x+5-5<2-5 x < -3 The values of x for which the inequality holds are exactly those x less than or equal to −3. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 25 25 of 71
26. 26. Inequality Property 2 Inequality Property 2 2A. If a < b and c is positive, then ac < bc. 2B. If a < b and c is negative, then ac > bc. Note: Inequality Property 2 also holds if < is replaced by >, < or >. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 26 26 of 71
27. 27. Example Inequality Property 2 Solve the inequality -3x + 1 > 7. Subtract 1 from both sides to isolate the x term on the left. -3x + 1 > 7 -3x + 1 - 1 > 7 - 1 -3x > 6 Divide by -3, or multiply by -1/3 to isolate the x. x < -2 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 27 27 of 71
28. 28. Standard Form of Linear Inequality A linear inequality of the form cx + dy < e can be written in the standard form 1. y < mx + b or y > mx + b if d ≠ 0, or 2. x < a or x > a if d = 0. Note: The inequality signs can be replaced by >, < or >. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 28 28 of 71
29. 29. Example Linear Inequality Standard Form Find the standard form of 5x - 3y < 6 and 4x > -8. (a) 5x - 3y < 6 (b) 4x > -8 5x - 3y < 6 -3y < - 5x + 6 y > (5/3)x - 2 4x > -8 x > -2 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 29 29 of 71
30. 30. Graph of x > a or x < a The graph of the inequality  x > a consists of all points to the right of and on the vertical line x = a;  x < a consists of all points to the left of and on the vertical line x = a. We will display the graph by crossing out the portion of the plane not a part of the solution. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 30 30 of 71
31. 31. Example Graph of x > a Graph the solution to 4x > -12. First write the equation in standard form. y 4x > -12 x = -3 x > -3 x Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 31 31 of 71
32. 32. Graph of y > mx + b or y < mx + b To graph the inequality, y > mx + b or y < mx + b: 1. Draw the graph of y = mx + b. 2. Throw away, that is, “cross out,” the portion of the plane not satisfying the inequality. The graph of y > mx + b consists of all points above or on the line. The graph of y < mx + b consists of all points below or on the line. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 32 32 of 71
33. 33. Example Graph of y > mx + b Graph the inequality 4x - 2y > 12. First write the equation in standard form. 4x - 2y > 12 y - 2y > - 4x + 12 y < 2x - 6 x y = 2x - 6 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 33 33 of 71
34. 34. Example Graph of System of Inequalities 2x 15 4x 2y 12 y Graph the system of inequalities 3y 0. The system in standard form is y y 2 x 3 2x y 5 6 0. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 34 34 of 71
35. 35. Summary Section 1.2 - Part 1  The direction of the inequality sign in an inequality is unchanged when a number is added to or subtracted from both sides of the inequality, or when both sides of the inequality are multiplied by the same positive number. The direction of the inequality sign is reversed when both sides of the inequality are multiplied by the same negative number. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 35 35 of 71
36. 36. Summary Section 1.2 - Part 2  The collection of points in the plane that satisfy the linear inequality ax + by < c or ax + by > c consists of all points on and to one side of the graph of the corresponding linear equation. After this inequality is put into standard form, the graph can be easily pictured by crossing out the half-plane consisting of the points that do not satisfy the inequality. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 36 36 of 71
37. 37. Summary Section 1.2 - Part 3 The feasible set of a system of linear inequalities (that is, the collection of points that satisfy all the inequalities) is best obtained by crossing out the points not satisfied by each inequality. The feasible set associated to the system of the previous example is a three-sided unbounded region. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 37 37 of 71
38. 38. Section 1.3 The Intersection Point of a Pair of Lines Copyright © 2014, 2010, 2007 Pearson Education, Inc. 38 of 71
39. 39. Outline 1. Solve y = mx + b and y = nx +c 2. Solve y = mx + b and x = a 3. Supply Curve 4. Demand Curve Copyright © 2014, 2010, 2007 Pearson Education, Inc. 39 of 71
40. 40. Solve y = mx + b and y = nx + c To determine the coordinates of the point of intersection of two lines y = mx + b and y = nx + c 1. Set y = mx + b = nx + c and solve for x. This is the x-coordinate of the point. 2. Substitute the value obtained for x into either equation and solve for y. This is the y-coordinate of the point. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 40 40 of 71
41. 41. Example Solve y = mx + b & y = nx + c Solve the system 2x 3y 7 4x 2y 9. Write the system in standard form, set equal and solve. y y y 2 x 3 2x 2 7 x 3 3 7 3 9 2 9 2x 2 8 41 x 3 6 41 x 16 41 9 y 2 16 2 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 41 5 8 41 of 71
42. 42. Example Point of Intersection Graph Point of Intersection: (41/16, 5/8) y y = 2x - 9/2 (41/16,5/8) x y = (-2/3)x + 7/3 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 42 42 of 71
43. 43. Solve y = mx + b and x = a To determine the coordinates of the point of intersection of two lines: y = mx + b and x = a 1. The x-coordinate of the point is x = a. 2. Substitute x = a into y = mx + b and solve for y. This is the y-coordinate of the point. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 43 43 of 71
44. 44. Example Solve y = mx + b & x = a Find the point of intersection of the lines y = 2x - 1 and x = 2. The x-coordinate of the point is x = 2. y Substitute x = 2 into y = 2x - 1 to get the y-coordinate. y = 2(2) - 1 = 3 Intersection Point: (2,3) y = 2x - 1 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 44 (2,3) x x=2 44 of 71
45. 45. Supply Curve p For every quantity q of a commodity, the supply curve specifies the price p that must be charged for a manufacturer to be willing to produce q units of the commodity. q Supply Curve Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 45 45 of 71
46. 46. Demand Curve p For every quantity q of a commodity, the demand curve gives the price p that must be charged in order for q units of the commodity to be sold. q Demand Curve Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 46 46 of 71
47. 47. Example Supply = Demand Suppose the supply and demand for a quantity is given by p = 0.0002q + 2 (p in dollars) and p = 0.0005q + 5.5. Determine both the quantity of the commodity that will be produced and the price at which it will sell when supply equals demand. p .0002q 2 .0005q 5.5 .0007q 3.5 q 5000 units p .0002(5000) 2 \$3 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 47 47 of 71
48. 48. Summary Section 1.3  The point of intersection of a pair of lines can be obtained by first converting the equations to standard form and then either equating the two expressions for y or substituting the value of x from the form x = a into the other equation. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 48 48 of 71
49. 49. Section 1.4 The Slope of a Straight Line Copyright © 2014, 2010, 2007 Pearson Education, Inc. 49 of 71
50. 50. Outline 1. Slope of y = mx + b 2. Geometric Definition of Slope 3. Steepness Property 4. Point-Slope Formula 5. Perpendicular Property 6. Parallel Property Copyright © 2014, 2010, 2007 Pearson Education, Inc. 50 of 71
51. 51. Slope of y = mx + b For the line given by the equation y = mx + b, the number m is called the slope of the line. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 51 51 of 71
52. 52. Example Slope of y = mx + b Find the slope. y = 6x - 9 m=6 y = -x + 4 m = -1 y=2 m=0 y=x m=1 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 52 52 of 71
53. 53. Geometric Definition of Slope Geometric Definition of Slope Let L be a line passing through the points (x1,y1) and (x2,y2) where x1 ≠ x2. Then the slope of L is given by the formula y2 y1 m . x2 x1 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 53 53 of 71
54. 54. Example Geometric Definition of Slope Use the geometric definition of slope to find the slope of y = 6x - 9. Let x = 0. Then y = 6(0) - 9 = -9. (x1,y1) = (0,-9) Let x = 2. Then y = 6(2) - 9 = 3. (x2,y2) = (2,3) 3 9 12 m 6 2 0 2 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 54 54 of 71
55. 55. Steepness Property Steepness Property Let the line L have slope m. If we start at any point on the line and move 1 unit to the right, then we must move m units vertically in order to return to the line. (Of course, if m is positive, then we move up; and if m is negative, we move down.) Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 55 55 of 71
56. 56. Example Steepness Property Use the steepness property to graph y = -4x + 3. The slope is m = -4. A point on the line is (0,3). If you move to the right 1 unit to x = 1, y must move down 4 units to y = 3 - 4 = -1. Copyright © 2014, 2010, 2007 Pearson Education, Inc. y (0,3) x (1,-1) y = -4x + 3 Slide 56 56 of 71
57. 57. Point-Slope Formula Point-Slope Formula The equation of the straight line through the point (x1,y1) and having slope m is given by y - y1 = m(x - x1). Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 57 57 of 71
58. 58. Example Point-Slope Formula Find the equation of the line that passes through (-1,4) with a slope of 3 . 5 Use the point-slope formula. 3 y 4 x 1 5 3 3 y 4 x 5 5 3 17 y x 5 5 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 58 58 of 71
59. 59. Perpendicular Property Perpendicular Property When two lines are perpendicular, their slopes are negative reciprocals of one another. That is, if two lines with slopes m and n are perpendicular to one another, then m = -1/n. Conversely, if two lines have slopes that are negative reciprocals of one another, they are perpendicular. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 59 59 of 71
60. 60. Example Perpendicular Property Find the equation of the line through the point (3,-5) that is perpendicular to the line whose equation is 2x + 4y = 7. The slope of the given line is -1/2. The slope of the desired line is -(-2/1) = 2. Therefore, y -(-5) = 2(x - 3) or y = 2x – 11. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 60 60 of 71
61. 61. Parallel Property Parallel Property Parallel lines have the same slope. Conversely, if two lines have the same slope, they are parallel. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 61 61 of 71
62. 62. Example Parallel Property Find the equation of the line through the point (3,-5) that is parallel to the line whose equation is 2x + 4y = 7. The slope of the given line is -1/2. The slope of the desired line is -1/2. Therefore, y -(-5) = (-1/2)(x - 3) or y = (-1/2)x - 7/2. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 62 62 of 71
63. 63. Graph of Perpendicular & Parallel Lines y = 2x - 11 2x + 4y = 7 y = (-1/2)x - 7/2 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 63 63 of 71
64. 64. Summary Section 1.4 - Part 1  The slope of the line y = mx + b is the number m. It is also the ratio of the difference between the y-coordinates and the difference between the x-coordinates of any pair of points on the line. The steepness property states that if we start at any point on a line of slope m and move 1 unit to the right, then we must move m units vertically to return to the line. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 64 64 of 71
65. 65. Summary Section 1.4 - Part 2  The point-slope formula states that the line of slope m passing through the point (x1, y1) has the equation y - y1 = m(x - x1).  Two lines are parallel if and only if they have the same slope. Two lines are perpendicular if and only if the product of their slopes is –1. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 65 65 of 71
66. 66. Section 1.5 The Method of Least Squares Copyright © 2014, 2010, 2007 Pearson Education, Inc. 66 of 71
67. 67. Outline 1. 2. 3. 4. Least Squares Problem Least Squares Error Least Squares Line Least Squares Using Technology Copyright © 2014, 2010, 2007 Pearson Education, Inc. 67 of 71
68. 68. Least Squares Problem Least Squares Problem Given observed data points (x1, y1), (x2, y2),…, (xN, yN) in the plane, find the straight line that “best” fits these points. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 68 68 of 71
69. 69. Least Squares Error Least Squares Error Let Ei be the vertical distance between the point (xi, yi) and the straight line. The least-squares error of the observed points with respect to this line is E = E12 + E22 +…+ EN2. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 69 69 of 71
70. 70. Example Least Squares Error Determine the least-squares error when the line y = 1.5x + 3 is used to approximate the data points (1,6), (4,5) and (6,14). Vertical Distance Ei2 (1, 4.5) 1.5 2.25 (4,9) 4 16 (6,12) 2 4 Data Point Point on Line (1,6) (4,5) (6,14) E = 22.25 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 70 70 of 71
71. 71. Graph of Least Squares Error (6,14) E3 (1,6) E1 E2 (4,5) Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 71 71 of 71
72. 72. Least Squares Line Least Squares Line Given observed data points (x1, y1), (x2, y2),…, (xN, yN) in the plane, the straight line y = mx + b for which the error E is as small as possible is determined by m N xy N b x y m N x 2 y x x 2 . Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 72 72 of 71
73. 73. Example Least Squares Error Find the least-squares line for the data points (1,6), (4,5) and (6,14). x y xy x2 1 6 6 1 4 5 20 16 6 14 84 36 x = 11 y = 25 xy = 110 x2 = 53 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 73 73 of 71
74. 74. Example Least Squares Error (2) 3 110 11 25 55 m 1.45 2 3 53 11 38 25 55 11 38 b 3.03 3 y 1.45 x 3.03 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 74 74 of 71
75. 75. Least Squares Using Technology Use Excel to find the least-squares line for the data points (1,6), (4,5) and (6,14). y = 1.4474x + 3.0263 Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 75 75 of 71
76. 76. Summary Section 1.5  The method of least squares finds the straight line that gives the best fit to a collection of points in the sense that the sum of the squares of the vertical distances from the points to the line is as small as possible. The slope and y-intercept of the least-squares line are usually found with formulae involving sums of coordinates or by using technology. Copyright © 2014, 2010, 2007 Pearson Education, Inc. Slide 76 76 of 71
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