932 CHAPTER 9 Systems and Matrices Nonlinear Systems with Nonreal Complex Solutions EXAMPLE 5 Solving a Nonlinear System (Nonreal Complex Solutions) Solve the system. x2 + y2 = 5 (1) 4x2 + 3y2 = 11 (2) SOLUTION Begin by eliminating a variable. -3x2 - 3y2 = -15 Multiply (1) by -3. 4x2 + 3y2 = 11 (2) x2 = -4 Add. x = {2-4 Square root property x = {2i 2-4 = i24 = 2i To find the corresponding values of y, substitute into equation (1). x2 + y2 = 5 (1) 12i22 + y2 = 5 Let x = 2i. -4 + y2 = 5 y2 = 9 y ={3 x2 + y2 = 5 (1) 1-2i22 + y2 = 5 Let x = -2i. -4 + y2 = 5 y2 = 9 y ={3 i 2 = -1 Checking the proposed solutions confirms the following solution set. 512i, 32, 12i, -32, 1-2i, 32, 1-2i, -326 Note that solutions with nonreal complex number components do not appear as intersection points on the graph of the system. S Now Try Exercise 43. An Application of Nonlinear Systems −6.6 −4.1 4.1 6.6 The graphs of the two equations in Example 5 do not intersect as seen here. The graphs are obtained by graphing y = {25 - x2 and y = {B11 - 4x2 3 . EXAMPLE 6 Using a Nonlinear System to Find Box Dimensions A box with an open top has a square base and four sides of equal height. The volume of the box is 75 in.3, and the surface area is 85 in.2. Find the dimensions of the box. SOLUTION Step 1 Read the problem. We must find the box width, length, and height. Step 2 Assign variables. Let x represent the length and width of the square base, and let y represent the height. See Figure 14. Step 3 Write a system of equations. Use the formula for the volume of a box, V = LWH, to write one equation using the given volume, 75 in.3. x2y = 75 Volume formula The surface consists of the base, whose area is x2, and four sides, each having area xy. The total surface area of 85 in.2 is used to write a second equation. x2 + 4xy = 85 Sum of areas of base and sides x x x x y y y y Figure 14
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