338 CHAPTER 6 Algebra, Graphs, and Functions The profit, p, of a company may depend on the amount of sales, s. The cost, c, of mailing a package may depend on the weight, w, of the package. Real-life problems, such as these two examples, often involve two or more variables. In this section, we will discuss how to graph equations with two variables. Graphing Linear Equations SECTION 6.6 LEARNING GOALS Upon completion of this section, you will be able to: 7 Understand the Cartesian coordinate system. 7 Graph linear equations by plotting points. 7 Graph linear equations by using the x- and y-intercepts. 7 Determine the slope of a line. 7 Graph linear equations by using the slope and y-intercept. Why This Is Important Many applications, including those described above, can be modeled using two variables. The relationship between two variables can be represented with a graph, and many times this graph is a straight line. In this section, we will encounter many real-life applications including those involving a company’s profits, simple interest, test grades, and retail spending on electronics. An understanding of how to graph lines is an important skill in many occupations. Cartesian Coordinate System To be able to work with equations with two variables requires understanding the Cartesian (or rectangular) coordinate system, named after the French mathematician René Descartes. The rectangular coordinate system consists of two perpendicular number lines (Fig. 6.3). The horizontal line is the x-axis, and the vertical line is the y-axis. The point of intersection of the x-axis and y-axis is called the origin. The axes divide the plane into four parts: the first, second, third, and fourth quadrants. 57. Grade of A To earn a grade of A in Philosophy, a student must have a 93% or greater average on four tests. Hyun currently has grades of 87%, 91%, and 98%. What range of grades on the fourth test would earn Hyun a grade of A in the class? 96% or higher 58. Passing Grade To earn a passing grade in Topology at Old College of Alabama, a student must have a 60% or greater average on 5 tests. Justice currently has grades of 61%, 75%, 57%, and 51%. What range of grades on the fifth test would earn Justice a passing grade? 56% or higher Challenge Problems/Group Activities 59. Painting a House Donovan is painting the exterior of his house. The instructions on the paint can indicate that 1 gal covers from 250 ft2 to 400 ft .2 The total surface of the house to be painted is 2750 ft .2 Determine the number of gallons of paint he could use and express the answer as an inequality. x 6.875 11 ≤ ≤ 60. Final Exam Teresa’s five test grades for the semester are 86%, 78%, 68%, 92%, and 72%. Her final exam counts one-third of her final grade. What range of grades on her final exam would result in Teresa receiving a final grade of B in the course? (See Example 11.) x 81.6 100, ≤ ≤ assuming 100 is the highest grade possible 61. A student multiplied both sides of the inequality x 4 1 3 − ≤ by 3− and forgot to reverse the direction of the inequality symbol. What is the relation between the student’s incorrect solution set and the correct solution set? Is there any number in both the correct solution set and the student’s incorrect solution set? If so, what is it? The student’s answer is x 12, ≤ − whereas the correct answer is x 12. ≥ − Yes, 12 − is in both solution sets. Research Activity 62. Inequalities Find an article that contains the mathematical concept of inequality. a) From the information in the article write a statement of inequality. b) Summarize the article and explain how you arrived at the inequality statement in part (a). Auremar /Shutterstock
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