942 CHAPTER 9 Systems and Matrices (b) Writing x … 3 as -3 … x … 3 shows that this inequality is satisfied by points in the region between and including x = -3 and x = 3. See Figure 20(a). The set of points that satisfies y … 0 includes the points below or on the x-axis. See Figure 20(b). Graph y = x + 1 and use a test point to verify that the solutions of y Ú x + 1 are on or above the boundary. See Figure 20(c). Because the solution sets of y … 0 and y Ú x + 1 shown in Figures 20(b) and (c) have no points in common, the solution set of the system is ∅. x 1 0 1 y y = 0 y ≤ 0 y = z x z + 1 y ≥ z x z + 1 x 1 0 1 y Figure 20 The solution set of the system is ∅ because there are no points common to all three regions simultaneously. x 1 0 1 y x = 3 z x z ≤ 3 x = –3 (a) (b) (c) S Now Try Exercises 43, 49, and 51. NOTE Although we gave three graphs in the solutions of Example 3, in practice we usually give only a final graph showing the solution set of the system. This would be an empty rectangular coordinate system in Example 3(b). Linear Programming One important application of mathematics to business and social science is linear programming. Linear programming is used to find an optimum value—for example, minimum cost or maximum profit. Procedures for solving linear programming problems were developed in 1947 by George Dantzig while he was working on a problem of allocating supplies for the Air Force in a way that minimized total cost. To solve a linear programming problem in general, use the following steps. (The italicized terms are defined in Example 4.) Solving a Linear Programming Problem Step 1 Write all necessary constraints and the objective function. Step 2 Graph the region of feasible solutions. Step 3 Identify all vertices (corner points). Step 4 Find the value of the objective function at each vertex. Step 5 The solution is given by the vertex producing the optimum value of the objective function. In this procedure, Step 5 is an application of the following theorem.
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