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- 1. Linear Equations in Two Variables<br />
- 2. Cartesian Coordinate System<br />The rectangular or Cartesian coordinate system consists of a horizontal number line, the x-axis, and a vertical number line, the y-axis. <br />The intersection of the axes is the origin. <br />
- 3. Cartesian Coordinate System<br />The axes divide the coordinate plane, or the xy-plane,into four regions called quadrants. <br />The quadrants are numbered counterclockwise and they do not include any points on the axes.<br />
- 4. Cartesian Coordinate System<br />Just as every real number corresponds to a point on the number line, every pair of real numbers corresponds to a point in the rectangular coordinate system. <br />Locating a point in the rectangular coordinate system that corresponds to a pair of real numbers is referred to as plotting or graphing the point.<br />
- 5. Cartesian Coordinate System<br />Graph the points corresponding to the following pairs:<br />(2, 4)<br />(4, 2)<br />(-2, -3)<br />(-1, 3)<br />(0, -4)<br />(4, -2)<br />
- 6. Cartesian Coordinate System<br /><ul><li>A pair of numbers, such as (2, 4), is called an ordered pair because the order of the numbers is important.
- 7. The pairs (4, 2) and (2, 4) correspond to different points.
- 8. The first number in an ordered pair is the x-coordinate and the second number is the y-coordinate.</li></li></ul><li>X - Intercept<br />An x-interceptis a point on the x-axis. <br />An x-intercept has a y-coordinate of 0.<br />Y - Intercept<br />A y-intercept is a point on the y-axis.<br />A y-intercept has an x-coordinate of 0.<br />
- 9. Exercises:<br />I. Plot the following points in a rectangular coordinate system. For each point, name the quadrant in which it lies or the axis on which it lies.<br />(2, 5) 3. (-5, 1) 5. (2, -6)<br />(-1, -6) 4. (0, 4) 6. (3, 0)<br />II.<br />Explain why the point (0, -2) is not located in Quadrant IV.<br />Explain why the point (-4, 0) is not located in Quadrant II.<br />
- 10. Distance and Midpoint Formulas<br />
- 11. The Distance Formula<br />The distance, d, between the points (x1, y1) and (x2,y2) in the rectangular coordinate system is<br />
- 12. Example<br />Find the distance between (-1, 2) and (4, -3).<br />Solution Letting (x1, y1) = (-1, 2) and (x2, y2) = (4, -3), we obtain<br />
- 13. The Midpoint Formula<br />Consider a line segment whose endpoints are (x1, y1) and (x2, y2). <br />To find the midpoint, take the average of the two x-coordinates and of the two y-coordinates.<br />The coordinates of the segment's midpoint are<br />
- 14. Example<br />Find the midpoint of the line segment with endpoints (1, -6) and (-8, -4).<br />Solution<br />(-7/2, -5) is midway between the points (1, -6) and (-8, -4).<br />
- 15. Exercises:<br />Find the midpoint and length of the line segment with the given endpoints. <br />(0, 0) and (6, 8)<br />(-1, 0) and (6, -2)<br />(-2, -5) and (5, 1)<br />(1/2, 7) and (5, 2/5)<br />(2, 4) and (0, 6)<br />(-3, 5) and (3, -3)<br />(0.2, 1.5) and (9.1, 6.4)<br />
- 16. Linear Equations in Two Variables<br />
- 17. Definition:<br />Let A, B, and C be real numbers such that A and B are not both zero. Then, an equation that can be written in the form:<br />Ax + By = C<br />is called a linear equation in two variables.<br />
- 18. Graphing a Linear Equations<br />
- 19. Definition:<br />A solution to a linear equation is an ordered pair (x, y) that makes the equation a true statement. <br />For y = 3x – 6, consider x = -1, 0,1, 2, 4.<br /> y = 3x – 6 ordered pair solutions<br />-9 = 3(-1) – 6 (-1, -9)<br />-6 = 3(0) – 6 (0, -6) <br />-3 = 3(1) – 6 (1, -3) <br />0 = 3(2) – 6 (2, -0)<br />1 = 3(4) – 6 (4, 1)<br />Therefore, the ordered pairs are solutions to the linear equation y = 3x – 6.<br />
- 20. Definition:<br />The Graph of an Equation in Two Variables<br />The graph of an equation in two variables is the graph of all ordered pair solutions to the equation.<br />
- 21. Graph: y=3x – 6. <br />
- 22. Graph: 3x + 2y = 12. <br />Solution:<br />Rewrite the equation, 3x +2y = 12<br /> 2y = -3x + 12<br /> y = -3/2 x + 6<br />Arbitrarily select some values for x and find the corresponding y-values. <br />Consider x = 0, 2, 4.<br />-3/2 (0) + 6 = 6 (0, 6)<br />-3/2 (2) + 6 = 3 (2, 3)<br />-3/2 (4) + 6 = 0 (4, 0)<br />
- 23. Graph: 3x + 2y = 12. <br />Solution:<br />Plot the points and draw a line through them.<br />(0, 6), <br />(2, 3), <br />(4, 0)<br />
- 24. Graph: y = -2. <br />Solution:<br />The equation y = -2 may be taken as y = 0x –2. <br />No matter what the value for x is, y will always be equal to -2. <br />The graph of such equation is a horizontal line passing through the point y = -2.<br />
- 25. Graph: x =1. <br />Solution:<br />The equation x = 1 may be taken as 0y + x = 1. <br />No matter what the value for y is, x will always equal to 1. <br /> The graph of such equation is a vertical line passing through the point x = 1 in this case.<br />
- 26. Using Intercepts for Graphing<br />If a line has distinct x- and y-intercepts, then they can be used as two points that determine the location of the line. <br />Since horizontal lines, vertical lines, and lines through the origin do not have two distinct intercepts, they cannot be graphed using only the intercepts.<br />
- 27. Example: Graph x + y = 5<br />Find the x – intercept: (x, 0)<br />y = 0: <br />x + (0) = 5<br />(5, 0)<br />Find the y – intercept: (0, y)<br />x = 0:<br />(0) + y = 5<br />(0, 5)<br />Graph the equation.<br />
- 28. Try this!<br />Graph the following: <br />y – x = 1<br />y = 2x<br />
- 29. Exercises:<br />Graph the following:<br />x + 2y = 4<br />2x - 3y = 6<br />x + 5 = 0<br />y + 1 = 0<br />6x + 3y = 0<br />
- 30. Slope of a Line<br />
- 31. Slope of a Line<br />The slope of a line is the ratio of the change in y-coordinate, or the rise, to the change in x-coordinate, or the run, between two points on the line.<br />
- 32. Finding the slope from a graph<br />Find the slope of the line by going from point A to point B.<br />A is located at (0, 3) and B at <br /> (2, 0). <br /> In going from A to B,<br /> change in y: -3 (going down)<br /> change in x: 2 (going right)<br /> So, the slope is<br /> m = -3/2<br />
- 33. Finding the slope from a graph<br />Find the slope of the line by going from point A to point B.<br />A is located at (2, 1) and B at <br /> (6, 3). <br /> In going from A to B,<br /> change in y: 2 (going up)<br /> change in x: 4 (going right)<br /> So, the slope is<br /> m = 2/4 = ½ <br />
- 34. Finding the slope from a graph<br />Find the slope of the line by going from point A to point B.<br />A is located at (0,0) and B at <br /> (-6, -3). <br /> In going from A to B,<br /> change in y: -3 (going down)<br /> change in x: -6 (going left)<br /> So, the slope is<br /> m = -3/-6 = ½ <br />
- 35. Finding the slope from coordinates<br />A is located at (0, 3) and B at (2, 0). <br />
- 36. Finding the slope from coordinates<br />A is located at (2, 1) and B at (6, 3). <br />
- 37. Finding the slope from coordinates<br />A is located at (0,0) and B at (-6, -3). <br />
- 38. Types of Slope<br />
- 39. Positive Slope<br />Lines that increase, or rise, from left to right have a positive slope.<br />
- 40. Negative Slope<br />Lines that decrease, or fall, from left to right have a negative slope.<br />
- 41. Zero Slope<br />Horizontal lines have a slope of zero.<br />
- 42. Undefined Slope<br />Vertical lines have an undefined slope.<br />
- 43. Exercises:<br />Find the slope of the line that contains each of the following points:<br />(2, 6) and (5, 1)<br />(-3, -1) and (4, 3)<br />(-2, -2) and (-1, 7)<br />(2, 6) and (1, 8)<br />(5, 1) and ( 5, -2)<br />(0, 3) and ( -1, 3) <br />(24.3, 11.9) and (3.57, 8.40)<br />(-2.7, 19.3), (5.46, -3.28)<br />
- 44. Equation of a Line<br />
- 45. Equation of a Line<br />
- 46. Example:<br />Find an equation for the line through (-2, 5) and slope -3.<br />Solution:<br />
- 47. Example:<br />Find the equation of the line through the given pair of points (3,5) and (4,7).<br />Solution:<br />Find the slope<br />Use the slope and one point, say (3, 5) in the point-slope form<br />
- 48. EXERCISES<br /> Find the slope of the line that passes through (2,3) and (4,3).<br />Find the slope and the y-intercept of 3x + 5y - 9 = 0.<br />What is the slope of the line y - 4x + 6 = 0?<br />What is the y-intercept of the line 3x + 2y = 5?<br />What is an equation of the line through (4,1) and (2,4) ?<br />What is an equation of the line passing through the points ( 6, -3 ) and ( -2, 3 )?<br />
- 49. EXERCISES<br />Find an equation of the line which has a slope of 2/3 and a y-intercept of 2?<br />Find an equation of the line with x-intercept -2 and y-intercept 2? <br />What is an equation of the line through (-8,1) with undefined slope ?<br />What is an equation of the line through (4,3) with slope equal to zero?<br />
- 50. EXERCISES<br />For each of the given equations, do the following:<br />Rewrite the equation in slope-intercept form<br />Determine the slope.<br />Find the intercepts (x and y).<br />Graph the equation.<br />y – 5x – 10 = 0<br />2y – x + 4 = 0<br />-2x + y + 8 = 0<br />3y + 2x + 6 = 0<br />5x – 5y – 15 = 0<br />
- 51. Finding Slope in an Application<br />
- 52. EXERCISES<br />Determine the slope of the ramp up the stairs.<br />Determine the slope of the aircraft’s takeoff path.<br />Determine the slope of the roof.<br />
- 53. Applications of Slope:<br />Rate of Change<br />
- 54. Example:<br />Worldwide carbon dioxide (CO2) emissions have increased from 14 billion tons in 1970 to 26 billion tons in 2000 (World Resources Institute, www.wri.org).<br />
- 55. Example:<br />Find and interpret the slope of the line.<br /> Solution:<br /> (1970, 14) and (2000, 26)<br /> CO2 emissions are increasing 0.4 billion tons per year.<br />
- 56. Example:<br />b) Use the slope to predict the amount of worldwide CO2 emissions in 2010.<br /> Solution:<br /> (1970, 14), (2000, 26), (2010, ?)<br />If the CO2 emissions keep increasing 0.4 billion tons per year, then in 10 years the level will go up 10(0.4) or 4 billion tons. <br />So in 2010 CO2 emissions will be 26 +4 = 30 billion tons.<br />
- 57. Let’s try this!<br />The list price for a new Ford Crown Victoria four-door sedan was $21,135 in 1998 and $24,513 in 2004 (www.edmunds.com).<br />a) Find the slope of the line shown in the accompanying figure.<br />b) Use the accompanying figure to predict the price in 2011.<br />c) Use the slope to predict the price in 2011.<br />

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