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= j 2 + ( 1) + ( j 2) + 5 = j 2 + j 2 + ( 1) + 5 =0+4 =4 Our third solution is (x , y) = ( j, 4)

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And finally Now it s time to check these solutions to be sure that they work I can t blame you if you re weary of doing arithmetic, and you d prefer to take these solutions on faith That s all right for now Tomorrow, when your mind is rested, come back and check them for complementary credit

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This is an open-book quiz You may (and should) refer to the text as you solve these problems Don t hurry! You ll find worked-out answers in App C The solutions in the appendix may not represent the only way a problem can be figured out If you think you can solve a particular problem in a quicker or better way than you see there, by all means try it! 1 Solve the following pair of equations as a two-by-two system, including the complexnumber solutions, if any Let x be the independent variable, and then define y as a function of x in each equation: 3x + y 1 = 0 and 2x 2 y + 1 = 0 2 Check the solution(s) to Prob 1 in the original equations for correctness 3 Solve the following pair of equations as a two-by-two system, including the complexnumber solutions, if any Let x be the independent variable, and then define y as a function of x in each equation: 3x + y 1 = 0 and 2x 2 3x y + 3 = 0 4 Check the solution(s) to Prob 3 in the original equations for correctness

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462 More Two-by-Two Systems

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5 Solve the following pair of equations as a two-by-two system, including the complexnumber solutions, if any Let x be the independent variable, and then define y as a function of x in each equation: x 2 + x y = 1 and x 2 2x y = 2 6 Check the solution(s) to Prob 5 in the original equations for correctness 7 Solve the following pair of equations as a two-by-two system, including the complexnumber solutions, if any Let x be the independent variable, and then define y as a function of x in each equation: x2 + y = 0 and 2x 3 y = 0 Explain why one of the solutions has multiplicity 2 8 Check the solution(s) to Prob 7 in the original equations for correctness 9 Solve the following pair of equations as a two-by-two system, including the complexnumber solutions, if any Let x be the independent variable, and then define y as a function of x in each equation: 4x 3 + 2x 2 + 2x 2y 8 = 0 and 3x 3 2x 2 + 4x y 5 = 0 What is the multiplicity of each solution 10 Check the solution(s) to Prob 9 in the original equations for correctness

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CHAPTER

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More Two-by-Two Graphs

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In this chapter, we ll graph the systems we solved in Chap 27 to see how they work in the realm of real numbers Real solutions always appear as intersection points between the graphs of functions However, when a two-by-two system has non-real complex solutions, those solutions don t show up as intersections between real-number graphs

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Linear and Quadratic

The first example we worked out in Chap 27 involved a system of two equations, one linear and the other quadratic: 2x y + 1 = 0 and y = x2 + x 5 We called x the independent variable, so we rewrote the first equation as a function of x The two-function system then became y = 2x + 1 and y = x2 + x 5

First, we tabulate some points The graphs we re about to create aren t meant to be precise We aren t trying to pinpoint the real solutions using the graphs; we ve already calculated them! The graphs serve only to geometrically illustrate the functions and their solutions