Ex 1: Solve a System of Nonlinear Equations (Substitution) — Transcript

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1. 0:00- WELCOME TO AN EXAMPLE ON HOW TO SOLVE A SYSTEM
2. 0:02OF NONLINEAR EQUATIONS.
3. 0:04WE WANT TO FIND ALL THE SOLUTIONS TO THE SYSTEM
4. 0:07Y = X SQUARED - 9 AND Y = 9 - X SQUARED.
5. 0:12SO WE'LL SOLVE THIS SYSTEM ALGEBRAICALLY
6. 0:14USING ELIMINATION OR SUBSTITUTION
7. 0:17AND THEN VERIFY OUR SOLUTION GRAPHICALLY.
8. 0:20BECAUSE BOTH OF OUR EQUATIONS ARE SOLVED FOR Y,
9. 0:23WE'LL GO AHEAD AND USE SUBSTITUTION.
10. 0:25SO LOOKING AT THE FIRST EQUATION, IF Y = X SQUARED - 9,
11. 0:30WE CAN SUBSTITUTE X SQUARED - 9 FOR Y IN THE SECOND EQUATION.
12. 0:35SO THIS WOULD GIVE US X SQUARED - 9 = 9 - X SQUARED.
13. 0:42AND NOW WE WANT TO SOLVE THIS EQUATION FOR X
14. 0:44AND WE'LL PERFORM BACK SUBSTITUTION
15. 0:47TO FIND THE VALUE OF Y.
16. 0:49SO WE FIRST WANT TO HAVE THE X SQUARED TERMS
17. 0:51ON ONE SIDE OF THE EQUATION,
18. 0:53SO WE'LL ADD X SQUARED TO BOTH SIDES.
19. 0:59THIS WILL GIVE US 1X SQUARED + 1X SQUARED
20. 1:01OR 2X SQUARED - 9 = 9.
21. 1:06THESE TWO TERMS ARE OPPOSITES, SO THEY SIMPLIFY OUT.
22. 1:10SO FOR THE NEXT STEP, WE'LL ADD 9 TO BOTH SIDES OF THE EQUATION.
23. 1:17THIS WOULD GIVE US 2X SQUARED,
24. 1:20THESE TWO TERMS SIMPLIFY OUT, = 18.
25. 1:24THE NEXT STEP, WE'LL DIVIDE BOTH SIDES BY 2.
26. 1:30SO NOW WE HAVE X SQUARED = 9.
27. 1:35NOW WE SHOULD RECOGNIZE FOR THIS EQUATION,
28. 1:37WE'LL HAVE TWO SOLUTIONS BECAUSE X IS SQUARED.
29. 1:41SO TO UNDO THE SQUARING ON X,
30. 1:42WE'LL TAKE THE SQUARE ROOT OF BOTH SIDES OF THE EQUATION,
31. 1:45AND THEN WE'LL ALSO INCLUDE A + OR - SIGN HERE TO MAKE SURE
32. 1:49WE OBTAIN BOTH SOLUTIONS.
33. 1:51SO WE HAVE X = + OR - 3 OR X = + 3 OR -3.
34. 1:59THIS IS NORMALLY HOW MOST TEXTS SHOW HOW TO SOLVE THIS EQUATION.
35. 2:03BUT THE REAL REASON WHY WE HAVE TWO SOLUTIONS HERE
36. 2:05IS BECAUSE WHEN WE TAKE THE SQUARE ROOT OF X SQUARED,
37. 2:08IT'S ACTUALLY EQUAL TO THE ABSOLUTE VALUE OF X,
38. 2:11BECAUSE THE RESULT MUST BE POSITIVE.
39. 2:13SO WE HAVE THE ABSOLUTE VALUE OF X = +3,
40. 2:17THEREFORE X CAN BE EITHER BOTH +3 OR -3.
41. 2:23SO NOW THAT WE KNOW X CAN BE EITHER + OR -3,
42. 2:26WE NEED TO PERFORM BACK SUBSTITUTION
43. 2:28INTO ONE OF THE ORIGINAL EQUATIONS
44. 2:29TO FIND THE CORRESPONDING Y VALUES.
45. 2:32LET'S GO AHEAD AND USE THE FIRST EQUATION WHERE WE HAVE
46. 2:34Y = X SQUARED - 9.
47. 2:38LET'S START WITH X = +3, SO WE WOULD HAVE Y = +3 SQUARED -9,
48. 2:48SO Y = 9 - 9 = 0 WHICH MEANS WHEN X = +3, Y IS 0.
49. 2:59BECAUSE X IS SQUARED WHEN X IS -3,
50. 3:04WE'LL HAVE THE SAME CORRESPONDING Y VALUE.
51. 3:06LET'S GO AHEAD AND SHOW IT.
52. 3:07IF X = -3, THEN WE HAVE Y = -3 SQUARED - 9,
53. 3:15BUT -3 SQUARED IS STILL 9, SO 9 - 9 = 0.
54. 3:21FOR OTHER SOLUTION IS X = -3 AND Y = 0.
55. 3:30SO THE SYSTEM OF NONLINEAR EQUATIONS HAS TWO SOLUTIONS
56. 3:34WHICH MEANS, IF YOU GRAPH THESE TWO EQUATIONS
57. 3:36ON THE COORDINATE PLANE,
58. 3:37THE TWO FUNCTIONS WHICH ARE PARABOLAS SHOULD INTERSECT
59. 3:41AT THESE TWO POINTS.
60. 3:42LET'S GO AHEAD AND VERIFY THAT.
61. 3:47NOTICE ONE PARABOLA OPENS UP AND ONE OPENS DOWN.
62. 3:50THERE ARE TWO POINTS OF INTERSECTIONS,
63. 3:52BOTH ON THE X-AXIS HERE AND HERE,
64. 3:57AND NOTICE THE COORDINATES OF THE POINT OF INTERSECTION
65. 4:00ARE (3,0) AND (-3,0) WHICH WERE THE SOLUTIONS
66. 4:05THAT WE FOUND ALGEBRAICALLY.
67. 4:08OKAY. I HOPE YOU FOUND THIS EXPLANATION HELPFUL.
68. 4:10THANK YOU FOR WATCHING.