340 CHAPTER 6 Algebra, Graphs, and Functions Graphing Linear Equations by Plotting Points Consider the following equation in two variables: y x 1. = + Every ordered pair that makes the equation a true statement is a solution to, or satisfies, the equation. We can mentally determine some ordered pairs that satisfy the equation y x 1 = + by picking some values of x and determining the corresponding values of y. For example, suppose we let x 1; = then y 1 1 2. = + = The ordered pair (1, 2) is a solution to the equation y x 1. = + We can make a chart of other ordered pairs that are solutions to the equation. Profile in Mathematics René Descartes According to legend, French mathematician and philosopher René Descartes (1596–1650) did some of his best thinking in bed. He was a sickly child, and so the Jesuits who undertook his education allowed him to stay in bed each morning as long as he liked. He carried this practice into adulthood, seldom getting up before noon. One morning as he watched a fly crawl about the ceiling, near the corner of his room, he was struck with the idea that the fly’s position could best be described by the connecting distances from it to the two adjacent walls. These became the coordinates of his rectangular coordinate system and were appropriately named after him (Cartesian coordinates) and not the fly. Solution A parallelogram is a figure that has opposite sides that are of equal length and are parallel. Parallel lines are two lines in the same plane that do not intersect. The horizontal distance between points B and C is 5 units (see Fig. 6.6). Therefore, the horizontal distance between points A and D must also be 5 units. This problem has two possible solutions, as illustrated in Fig. 6.6. In each figure, we have indicated the given points in red. B(2, 4) C(7, 4) D(6, 2) 5 units 5 units A(1, 2) (a) y 2221 1 2 3 4 5 6 7 4 5 6 3 2 1 21 (b) y 4 5 6 1 21 x 2221 2423 1 2 3 4 5 6 7 x D(24, 2) B(2, 4) 5 units C(7, 4) A(1, 2) 5 units 2 3 Figure 6.6 The solutions are the points (6, 2) and ( 4, 2). − 7 Now try Exercise 91 x y Ordered Pair 1 2 (1, 2) 2 3 (2, 3) 3 4 (3, 4) 4.5 5.5 (4.5, 5.5) 3− 2− ( 3, 2) − − How many other ordered pairs satisfy the equation? Infinitely many ordered pairs satisfy the equation. Since we cannot list all the solutions, we show them by means of a graph. A graph is an illustration of all the points whose coordinates satisfy an equation. The points (1, 2), (2, 3), (3, 4), (4.5, 5.5), and ( 3, 2) − − are plotted in Fig. 6.7. With a straightedge we can draw one line that contains all these points. This line, when extended indefinitely in both directions, passes through all the points in the plane that satisfy the equation y x 1. = + The arrows on the ends of the line indicate that the line extends indefinitely.
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