Chapter 1. Linear Equations and Straight Lines. 2 of 71. Copyright 2014, 2010, 2007 Pearson Education, Inc.

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1 Chapter 1 Linear Equations and Straight Lines 2 of 71

2 Outline 1.1 Coordinate Systems and Graphs 1.4 The Slope of a Straight Line 1.3 The Intersection Point of a Pair of Lines 1.2 Linear Inequalities 1.5 The Method of Least Squares 3 of 71

3 Section 1.1 Coordinate Systems and Graphs 4 of 71

4 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. Slide 5 5 of 71

5 Example Coordinate Line Graph the points -3/5, 1/2 and 15/8 on a coordinate line. -3/5 1/2 15/ Origin Unit length Negative numbers Positive numbers Slide 6 6 of 71

6 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 y O y-axis x-axis x Slide 7 7 of 71

7 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 x-axis. The second tells the number of units from the point to the y-axis. Slide 8 8 of 71

8 Example Coordinate Plane Plot the points: (2,1), (-1,3), (-2,-1) and (0,-3). (-1,3) -1 y -1 (-1,-2) (2,1) 1 x (0,-3) Slide 9 9 of 71

9 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. Slide of 71

10 Example Graph of an Equation Sketch the graph of the equation y = 2x - 1. x y = 2x - 1 (x,y) -2 2(-2) - 1 = -5 (-2,-5) -1 2(-1) - 1 = -3 (-1,-3) 0 2(0) - 1 = -1 (0,-1) 1 2(1) - 1 = 1 (1,1) 2 2(2) - 1 = 3 (2,3) (1,1) (0,-1) (-1,-3) (-2,-5) y (2,3) x Slide of 71

11 General Linear Equation An equation that can be written in the form cx + dy = e (c, d, e constants) is called a linear equation in x and y. Slide of 71

12 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. Slide of 71

13 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 Slide of 71

14 Graph of x = a The equation x = a graphs into a vertical line a units from the y-axis. y x = 2 x = -3 y x x Slide of 71

15 Intercepts x-intercept: the point where the graph intersects the x-axis. This corresponds to a point on the graph that has a y-coordinate of 0. Similarly y-intercept: the point where the graph intersects the y-axis. This corresponds to a point on the graph that has a x-coordinate of 0. Slide of 71

16 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. Slide of 71

17 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 y-intercept: Let x = 0 y = 2(0) - 1 = -1 y = 2x - 1 (0,-1) (1/2,0) x Slide of 71

18 Summary Section 1.1 Ø Cartesian coordinate systems associate a number with each point of a line and associate a pair of numbers with each point of a plane. Ø The collection of points in the plane that satisfy the equation ax + by = c lies on a straight line. Ø After this equation is put into one of the standard forms y = mx + b or x = a, the graph is easily drawn. Slide of 71

19 Section 1.4 The Slope of a Straight Line 20 of 71

20 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. Slide of 71

21 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 Slide of 71

22 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 y2 y1 m =. x x 2 1 Slide of 71

23 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) m 3 ( 9) 12 = = = Slide of 71

24 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.) Slide of 71

25 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 y (0,3) (1,-1) x down 4 units to y = 3-4 = -1. y = -4x + 3 Slide of 71

26 Point-Slope Formula Point-Slope Formula The 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 ). Slide of 71

27 Example Point-Slope Formula Find the equation of the line that passes through (-1,4) with a slope of 3. Use the point-slope formula. 3 y 4= y 4 = x y = x ( x ( )) Slide of 71

28 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. Slide of 71

29 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. Slide of 71

30 Parallel Property Parallel Property Parallel lines have the same slope. Conversely, if two lines have the same slope, they are parallel. Slide of 71

31 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. Slide of 71

32 Graph of Perpendicular & Parallel Lines 2x + 4y = 7 y = 2x - 11 y = (-1/2)x - 7/2 Slide of 71

33 Summary Section Part 1 Ø 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. Ø 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. Slide of 71

34 Summary Section Part 2 Ø 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 ). Ø 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. Slide of 71

35 Section 1.3 The Intersection Point of a Pair of Lines 36 of 71

36 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. Slide of 71

37 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 = x y = 2x _ y = x+ = 2x x = x = y = 2. _ = Slide of 71

38 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 Slide of 71

39 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. Slide of 71

40 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 Intersection Point: (2,3) y = 2x - 1 y (2,3) x = 2 x Slide of 71

41 Summary Section 1.3 Ø 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. Slide of 71

42 Section 1.2 Linear Inequalities 43 of 71

43 Definitions of Inequality Signs Ø a < b means a lies to the left of b on the number line (if the number line is the x-axis) or a lies below b on the number line (if the number line is the y-axis). Ø a < b means a = b or a < b. Ø Similarly, a > b means a lies to the right of b or above b on the number line (depending on the axis). Ø a > b means a = b or a > b. Slide of 71

44 Inequality Signs Example Which of the following statements are true? 1 < 4 True -1 > -4 True 2 < 3 True 0 < -2 False 3 > 3 True Slide of 71

45 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 >. Slide of 71

46 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 < 2-5 x < -3 The values of x for which the inequality holds are exactly those x less than or equal to 3. Slide of 71

47 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 >. Slide of 71

48 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 > x > 6 Divide by -3, or multiply by -1/3 to isolate the x. x < -2 Slide of 71

49 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 >. Slide of 71

50 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 Slide of 71

51 Graph of x > a or x < a The graph of the inequality Ø x > a consists of all points to the right of and on the vertical line x = a; Ø x < a consists of all points to the left of and on the vertical line x = a. Ø We will display the graph by crossing out the portion of the plane not a part of the solution. Slide of 71

52 Example Graph of x > a Graph the solution to 4x > -12. First write the equation in standard form. 4x > -12 x > -3 y x = -3 x Slide of 71

53 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. 3. 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. Slide of 71

54 Example Graph of y > mx + b Graph the inequality 4x - 2y > 12. First write the equation in standard form. 4x - 2y > 12-2y > - 4x + 12 y < 2x - 6 y x y = 2x - 6 Slide of 71

55 Example Graph of System of Inequalities Graph the system of inequalities The system in standard form is y y 2 x x 6 y 0. y < -2/3 x + 5 2x + 3y 15 _ 4x 2y 12 y 0. y > 2 x - 6 Feasible set y < 0 Slide of 71

56 Summary Section Part 1 Ø 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. Ø The direction of the inequality sign is reversed when both sides of the inequality are multiplied by the same negative number. Slide of 71

57 Summary Section Part 2 Ø 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. Ø 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. Slide of 71

58 Summary Section Part 3 Ø 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. Slide of 71

SNAP Centre Workshop. Graphing Lines

SNAP Centre Workshop. Graphing Lines SNAP Centre Workshop Graphing Lines 45 Graphing a Line Using Test Values A simple way to linear equation involves finding test values, plotting the points on a coordinate plane, and connecting the points.