1. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 1 of 71 Chapter 1 Linear Equations and Straight Lines

2. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 2 of 71 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

3. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 3 of 71 1.1 Coordinate Systems and Graphs 1.Coordinate Line 2.Coordinate Plane 3.Graph of an Equation 4.Linear Equation 5.Standard Form of Linear Equation 6.Graph of x = a 7.Intercepts 8.Graph of y = mx + b

4. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 4 of 71 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.

5. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 5 of 71 Example Coordinate Line Graph the points -3/5, 1/2 and 15/8 on a coordinate line. -4 -3 -2 -1 0 1 2 3 4 Origin -3/5 1/2 15/8 Unit length Positive numbers Negative numbers

6. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 6 of 71 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. Origin O x-axis y-axis x y

7. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 7 of 71 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.

8. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 8 of 71 Example Coordinate Plane Plot the points: (2,1), (-1,3), (-2,-1) and (0,-3). x y (2,1) 2 1 (0,-3) -3 3 (-1,3) -2 (-1,-2)

9. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 9 of 71 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.

10. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 10 of 71 Example Graph of an Equation Sketch the graph of the equation y = 2x - 1. x y -22(-2) - 1 = -5 2(-1) - 1 = -3 02(0) - 1 = -1 12(1) - 1 = 1 22(2) - 1 = 3 x y (-2,-5) (-1,-3) (0,-1) (1,1) (2,3)

11. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 11 of 71 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.

12. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 12 of 71 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.

13. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 13 of 71 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

14. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 14 of 71 Graph of x = a The equation x = a graphs into a vertical line a units from the y-axis. x y x = 2 x y x = -3

15. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 15 of 71 Intercepts x-intercept: a point on the graph that has a y- coordinate of 0. This corresponds to a point where the graph intersects the x-axis. y-intercept: the point on the graph that has a x- coordinate of 0. This corresponds to the point where the graph intersects the y-axis.

16. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 16 of 71 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.

17. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 17 of 71 Example Graph of Linear Equation Use the intercepts to graph y = 2x - 1. x-intercept: Let y = 0 0 = 2x - 1 x = 1/2 y-intercept: Let x = 0 y = 2(0) - 1 = -1 x y (1/2,0) (0,-1) y = 2x - 1

18. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 18 of 71 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.

19. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 19 of 71 1.2 Linear Inequalities 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

20. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 20 of 71 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.

21. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 21 of 71 Which of the following statements are true? 1 < 4 -1 > -4 2 < 3 0 > -2 3 > 3 Inequality Signs Example -4 -3 -2 -1 0 1 2 3 4 True

22. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 22 of 71 Inequality Property 1 Inequality Property 1Suppose 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,.

23. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 23 of 71 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

24. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 24 of 71 Inequality Property 2 2A.If a < b and c is positive, then ac < bc. 2B.If a bc. Note: Inequality Property 2 also holds if,.

25. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 25 of 71 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 to isolate the x. x < -2

26. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 26 of 71 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 if d ≠ 0, or 2. x a if d = 0. Note: The inequality signs can be replaced by >,.

27. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 27 of 71 Example Linear Inequality Standard Form Find the standard form of 5x - 3y -8. (a) 5x - 3y < 6(b) 4x > -8 5x - 3y < 6 -3y < - 5x + 6 y > (5/3)x - 2 4x > -8 x > -2

28. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 28 of 71 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.

29. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 29 of 71 Example Graph of x > a Graph the solution to 4x > -12. First write the equation in standard form. 4x > -12 x > -3 x y x = -3

30. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 30 of 71 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.

31. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 31 of 71 y = 2x - 6 Graph the inequality 4x - 2y > 12. First write the equation in standard form. 4x - 2y > 12 - 2y > - 4x + 12 y < 2x - 6 Example Graph of y > mx + b x y

32. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 32 of 71 Example Graph of System of Inequalities Graph the system of inequalities The system in standard form is

33. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 33 of 71 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.

34. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 34 of 71 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.

35. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 35 of 71 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.

36. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 36 of 71 1.3 The Intersection Point of a Pair of Lines 1.Solve y = mx + b and y = nx + c 2.Solve y = mx + b and x = a 3.Supply Curve 4.Demand Curve

37. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 37 of 71 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.

38. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 38 of 71 Example Solve y = mx + b & y = nx + c Solve the system Write the system in standard form, set equal and solve.

39. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 39 of 71 Example Point of Intersection Graph Point of Intersection: (41/16, 5/8) x y y = 2x - 9/2 y = (-2/3)x + 7/3 (41/16,5/8)

40. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 40 of 71 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.

41. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 41 of 71 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. Substitute x = 2 into y = 2x - 1 to get the y-coordinate. y = 2(2) - 1 = 3 Point: (2,3) x y y = 2x - 1 (2,3) x = 2

42. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 42 of 71 Supply Curve 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 p Supply Curve

43. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 43 of 71 Demand Curve 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 p Demand Curve

44. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 44 of 71 Example Supply = Demand Suppose the supply and demand for a quantity is given by p = 0.0002q + 2 (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.

45. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 45 of 71 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.

46. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 46 of 71 1.4 The Slope of a Straight Line 1.Slope of y = mx + b 2.Geometric Definition of Slope 3.Steepness Property 4.Point-Slope Formula 5.Perpendicular Property 6.Parallel Property

47. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 47 of 71 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.

48. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 48 of 71 Example Slope of y = mx + b Find the slope. y = 6x - 9 y = -x + 4 y = 2 y = x m = 6 m = -1 m = 0 m = 1

49. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 49 of 71 Geometric Definition of Slope Geometric Definition of Slope Let L be a line passing through the points (x 1,y 1 ) and (x 2,y 2 ) where x 1 ≠ x 2. Then the slope of L is given by the formula

50. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 50 of 71 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. (x 1,y 1 ) = (0,-9) Let x = 2. Then y = 6(2) - 9 = 3. (x 2,y 2 ) = (2,3)

51. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 51 of 71 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.)

52. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 52 of 71 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. y = -4x + 3 (0,3) x y (1,-1)

53. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 53 of 71 Point-Slope Formula Point-Slope FormulaThe equation of the straight line through the point (x 1,y 1 ) and having slope m is given by y - y 1 = m(x - x 1 ).

54. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 54 of 71 Example Point-Slope Formula Find the equation of the line through the point (-1,4) with a slope of. Use the point-slope formula.

55. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 55 of 71 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.

56. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 56 of 71 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.

57. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 57 of 71 Parallel Property Parallel Property Parallel lines have the same slope. Conversely, if two lines have the same slope, they are parallel.

58. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 58 of 71 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.

59. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 59 of 71 Graph of Perpendicular & Parallel Lines 2x + 4y = 7 y = (-1/2)x - 7/2 y = 2x - 11

60. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 60 of 71 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.

61. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 61 of 71 Summary Section 1.4 - Part 2  The point-slope formula states that the line of slope m passing through the point (x 1, y 1 ) has the equation y - y 1 = m(x - x 1 ).  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.

62. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 62 of 71 1.5 The Method of Least Squares 1.Least Squares Problem 2.Least Squares Error 3.Least Squares Line 4.Least Squares Using Technology

63. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 63 of 71 Least Squares Problem Least Squares Problem Given observed data points (x 1, y 1 ), (x 2, y 2 ),…, (x N, y N ) in the plane, find the straight line that “best” fits these points.

64. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 64 of 71 Least Squares Error Least Squares Error Let E i be the vertical distance between the point (x i, y i ) and the straight line. The least-squares error of the observed points with respect to this line is E = E 1 2 + E 2 2 +…+ E N 2.

65. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 65 of 71 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). Data PointPoint on LineVertical DistanceEi2Ei2 (1,6) (1, 4.5)1.52.25 (4,5) (4,9)416 (6,14) (6,12)24 E =22.25

66. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 66 of 71 Graph of Least Squares Error (1,6) (4,5) (6,14) E1E1 E2E2 E3E3

67. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 67 of 71 Least Squares Line Least Squares Line Given observed data points (x 1, y 1 ), (x 2, y 2 ),…, (x N, y N ) in the plane, the straight line y = mx + b for which the error E is as small as possible is determined by

68. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 68 of 71 Example Least Squares Error Find the least-squares line for the data points (1,6), (4,5) and (6,14). xyxyx2x2 1661 452016 6148436 x = 11 y = 25 xy = 110 x 2 = 53

69. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 69 of 71 Example Least Squares Error (2)

70. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 70 of 71 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

71. Finite Mathematics & Its Applications, 10/e by Goldstein/Schneider/SiegelCopyright © 2010 Pearson Education, Inc. 71 of 71 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.