SECTION 11.6 Systems of Nonlinear Equations 827 ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 11.6 Assess Your Understanding 1. Graph the equation: = + y x3 2 (pp. 37–38) 2. Graph the equation: + = y x 4 2 (pp. 682–685) 3. Graph the equation: = − y x 1 2 2 (pp. 705–712) 4. Graph the equation: + = x y4 4 2 2 (pp. 692–697) Historical Feature In the beginning of this section, it was stated that imagination and experience are important in solving systems of nonlinear equations. Indeed, these kinds of problems lead into some of the deepest and most difficult parts of modern mathematics. Look again at the graphs in Examples 1 and 2 of this section (Figures 19 and 21). Example 1 has two solutions, and Example 2 has four solutions. We might conjecture that the number of solutions is equal to the product of the degrees of the equations involved. This conjecture was made by Étienne Bézout (1730 —1783), but working out the details took about 150 years. It turns out that arriving at the correct number of intersections requires counting not only the complex number intersections, but also those intersections that, in a certain sense, lie at infinity. For example, a parabola and a line lying on the axis of the parabola intersect at the vertex and at infinity. This topic is part of the study of algebraic geometry. A papyrus dating back to 1950 BC contains the following problem: “A given surface area of 100 units of area shall be represented as the sum of two squares whose sides are to each other as 1 is to 3 4 . ” Solve for the sides by solving the system of equations + = = ⎧ ⎨ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪⎪ x y x y 100 3 4 2 2 Historical Problems = x 6 units, = y 8 units Skill Building In Problems 5–24, graph each equation of the system. Then solve the system to find the points of intersection. 5. = + = + ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ y x y x 1 1 2 6. = + = + ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ y x y x 1 4 1 2 7. = − = − ⎧ ⎨ ⎪⎪⎪ ⎩ ⎪⎪ ⎪ y x y x 36 8 2 8. = − = + ⎧ ⎨ ⎪⎪⎪ ⎩ ⎪⎪ ⎪ y x y x 4 2 4 2 9. = = − ⎧ ⎨ ⎪⎪⎪ ⎩ ⎪⎪⎪ y x y x 2 10. = = − ⎧ ⎨ ⎪⎪⎪ ⎩ ⎪⎪⎪ y x y x 6 11. = = − ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ x y x y y 2 2 2 12. = − = − + ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ y x y x x 1 6 9 2 13. + = + + = ⎧ ⎨ ⎪⎪⎪ ⎩ ⎪⎪⎪ x y x x y 4 2 0 2 2 2 2 14. + = + + = ⎧ ⎨ ⎪⎪⎪ ⎩ ⎪⎪⎪ x y x y y 8 4 0 2 2 2 2 15. = − + = ⎧ ⎨ ⎪ ⎩⎪⎪ y x x y 3 5 5 2 2 16. + = = + ⎧ ⎨ ⎪ ⎩⎪⎪ x y y x 10 2 2 2 17. + = − = ⎧ ⎨ ⎪⎪⎪ ⎩ ⎪⎪⎪ x y y x 4 4 2 2 2 18. + = − = ⎧ ⎨ ⎪⎪⎪ ⎩ ⎪⎪⎪ x y x y 16 2 8 2 2 2 19. = + = ⎧ ⎨ ⎪ ⎩⎪⎪ xy x y 4 8 2 2 20. = = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ x y xy 1 2 21. + = = − ⎧ ⎨ ⎪ ⎩⎪⎪ x y y x 4 9 2 2 2 22. { = = + xy y x 1 2 1 23. = − = − ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ y x y x 4 6 13 2 24. + = = ⎧ ⎨ ⎪ ⎩⎪⎪ x y xy 10 3 2 2 In Problems 25–54, solve each system. Use any method you wish. 25. + = = ⎧ ⎨ ⎪ ⎩⎪⎪ x y xy 2 18 4 2 2 26. − = + = ⎧ ⎨ ⎪⎪⎪ ⎩ ⎪⎪⎪ x y x y 21 7 2 2 27. = + + = ⎧ ⎨ ⎪ ⎩⎪⎪ y x x y 3 2 3 4 2 2 28. − = − = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ x y y x 4 16 2 2 2 2 29. + + = + + − = − ⎧ ⎨ ⎪ ⎩⎪⎪ x y x y y x 1 0 6 5 2 2 30. − + = = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ x xy y xy 2 8 4 2 2 31. − + = + = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ x xy y x y 9 8 4 70 3 2 10 2 2 32. − + + + = − + = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ y xy y x x y 2 3 6 2 4 0 2 3 4 0 2 33. − + = + = ⎧ ⎨ ⎪ ⎩⎪⎪ x y x y 4 7 0 3 31 2 2 2 2 34. − + = − + = ⎧ ⎨ ⎪⎪⎪ ⎩ ⎪⎪⎪ x y x y 3 2 5 0 2 2 0 2 2 2 2 35. − + = + = ⎧ ⎨ ⎪ ⎩⎪⎪ x y x y 7 3 5 0 3 5 12 2 2 2 2 36. − + = − + = ⎧ ⎨ ⎪⎪⎪ ⎩ ⎪⎪⎪ x y x y 3 1 0 2 7 5 0 2 2 2 2 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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