Solving Systems of Equations using Elimination — Transcript

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1. 0:01- TODAY WE'LL DISCUSS THE THIRD METHOD
2. 0:02FOR SOLVING SYSTEMS OF EQUATIONS,
3. 0:05AND THAT METHOD WILL BE THE ELIMINATION OR ADDITION METHOD.
4. 0:10LET'S TAKE A LOOK AT HOW THIS IS GOING TO WORK.
5. 0:12ESSENTIALLY WHAT WE'RE GOING TO DO
6. 0:13IS ADD THESE EQUATIONS TOGETHER.
7. 0:16THE REASON WE'RE ALLOWED TO DO THAT IS BECAUSE
8. 0:18IF WE TAKE A LOOK AT THE SECOND EQUATION,
9. 0:20SINCE THIS EXPRESSION IS EQUAL TO 7,
10. 0:22WE COULD ADD THIS EXPRESSION TO THE LEFT SIDE OF THIS EQUATION
11. 0:26AND ADD 7 TO THE RIGHT SIDE OF THIS EQUATION
12. 0:28SINCE THEY ARE EQUAL.
13. 0:30NOW LET'S SEE WHAT HAPPENS WHEN WE DO THIS.
14. 0:33IF WE ADD THE X TERMS TOGETHER WE'D HAVE 5X.
15. 0:37NOW, WHEN WE ADD THE Y TERMS TOGETHER SINCE THEY'RE OPPOSITES
16. 0:40WE GET 0, SO THESE TERMS ARE ELIMINATED.
17. 0:45AND ON THE RIGHT, 8 + 7 WOULD GIVE US 15.
18. 0:48WE HAVE AN EQUATION WITH ONE VARIABLE.
19. 0:50WE CAN SOLVE THAT DIVIDING BOTH SIDES BY 5.
20. 0:53WE HAVE X = 3.
21. 0:56REMEMBER, ONE SOLUTION CONSISTS OF AN X AND A Y VALUE.
22. 1:00WE KNOW THE X VALUE IS 3, BUT WE STILL HAVE TO FIND THE Y-VALUE.
23. 1:04WHAT WE DO NOW IS WE REPLACE X WITH 3 IN EITHER EQUATION
24. 1:08AND SOLVE FOR Y.
25. 1:10I WILL GO AHEAD AND USE THE FIRST EQUATION FOR THIS PURPOSE.
26. 1:14SO WE KNOW THAT X IS 3, SO WE'D HAVE 4 x 3 + 3Y = 8.
27. 1:21SO WE'LL SIMPLIFY AND SOLVE.
28. 1:23- 12 ON BOTH SIDES.
29. 1:263Y = -4.
30. 1:30DIVIDE BOTH SIDES BY 3.
31. 1:32Y = -4/3.
32. 1:36WE HAVE ONE SOLUTION FOR THIS SYSTEM.
33. 1:38THEREFORE, THE SYSTEM IS CONSISTENT AND INDEPENDENT.
34. 1:42YOU CAN SEE THIS METHOD WORKS VERY NICELY.
35. 1:44IT'S A LOT FASTER THAN TRYING TO GRAPH THESE TWO LINEAR EQUATIONS
36. 1:48AND ESTIMATE A POINT OF INTERSECTION.
37. 1:51LET'S TAKE A LOOK AT IN GENERAL HOW WE'RE GOING TO SOLVE
38. 1:54A SYSTEM USING THE ELIMINATION METHOD.
39. 1:57STEP ONE, WE'LL MULTIPLY ONE OR BOTH OF THE EQUATIONS
40. 2:00BY THE APPROPRIATE NUMBERS
41. 2:02SO THAT ONE OF THE VARIABLES WILL HAVE THE SAME COEFFICIENT
42. 2:05WITH OPPOSITE SIGNS.
43. 2:07THEN WE'LL ADD THE TWO EQUATIONS TOGETHER
44. 2:09BECAUSE ONE OF THE VARIABLES HAD THE SAME COEFFICIENT
45. 2:12WITH THE OPPOSITE SIGN AND WILL BE ELIMINATED
46. 2:14WHEN ADDED TO THE OTHER EQUATION.
47. 2:16THEN WE'LL SOLVE THE RESULTING EQUATION
48. 2:19AND THEN SUBSTITUTE THIS ANSWER BACK
49. 2:20INTO ONE OF THE ORIGINAL EQUATIONS
50. 2:22TO FIND THE REMAINING VARIABLE.
51. 2:24SO THE KEY HERE, IF WE WANT TO SOLVE THIS USING ELIMINATION,
52. 2:26EITHER THE X TERMS OR THE Y TERMS MUST BE OPPOSITES.
53. 2:30SINCE THE X TERMS ARE ALREADY OPPOSITE SIGNS,
54. 2:33I'M GOING TO GO AHEAD AND TRY TO ELIMINATE THE X TERMS.
55. 2:37NOW, THE LEAST COMMON MULTIPLE OF 2 AND 3 WOULD BE 6,
56. 2:41SO I'M GOING TO CONVERT THIS TO A 6X AND THIS TO A -6X.
57. 2:49THE WAY I'M GOING TO DO THAT
58. 2:50IS I'M GOING TO TAKE THIS FIRST EQUATION AND MULTIPLY BY A 2,
59. 2:55AND I'LL MULTIPLY THE SECOND EQUATION BY 3.
60. 2:59WHEN I DO THIS, I DO HAVE TO MULTIPLY THE ENTIRE EQUATION
61. 3:01BY 2 TO MAINTAIN THE EQUALITY, SO WE'D HAVE 6X + 10Y = 8.
62. 3:08HERE WE WOULD HAVE -6X + 9Y = 30.
63. 3:14NOW WE WILL ADD THE 2 EQUATIONS TOGETHER.
64. 3:16THAT'S SOMETIMES WHY IT IS CALLED
65. 3:18THE ADDITION METHOD.
66. 3:19OF COURSE, WHEN WE ADD A 6 AND A -6 THAT WOULD BE 0.
67. 3:23SO THE RESULT WOULD BE 19Y IS EQUAL TO 38.
68. 3:28DIVIDE BY 19.
69. 3:30WE HAVE A Y VALUE EQUAL TO 2.
70. 3:33WE'RE GOING TO TAKE THIS VALUE,
71. 3:34SUB IT INTO ONE OF THESE EQUATIONS AND THEN SOLVE FOR X.
72. 3:38I'LL GO AHEAD AND JUST USE THE FIRST EQUATION. WE HAVE 3X + 5Y,
73. 3:42BUT WE NOW KNOW Y IS 2, SO THAT WOULD BE A 10 = 4.
74. 3:48SUBTRACTING 10 ON BOTH SIDES, DIVIDING BY 3 WE HAVE X = -2.
75. 3:54THEREFORE, OUR SOLUTION TO THIS SYSTEM WOULD BE X = -2, Y = 2,
76. 4:00SO WE CAN CLASSIFY THIS SYSTEM AS CONSISTENT AND INDEPENDENT.
77. 4:05LET'S TAKE A LOOK AT ANOTHER.
78. 4:07THIS TIME I'LL TRY TO ELIMINATE THE Y TERMS.
79. 4:10SINCE WE HAVE A -3Y AND 6Y,
80. 4:14IF WE CAN MAKE THIS A -6Y THEY'D BE OPPOSITES.
81. 4:18SO IN THIS CASE WE ONLY HAVE TO MULTIPLY
82. 4:19THE FIRST EQUATION BY 2.
83. 4:22WE CAN LEAVE THE SECOND EQUATION EXACTLY THE SAME,
84. 4:25SO THE RESULT WOULD BE 4X - 6Y = -2.
85. 4:31SECOND EQUATION REMAINS THE SAME.
86. 4:33NOW, WHEN WE TRY TO ADD THESE TWO EQUATIONS TOGETHER,
87. 4:36NOTICE HOW THE X TERMS AND THE Y TERMS ARE OPPOSITES.
88. 4:40SO ON THE LEFT SIDE OF THIS EQUATION WE'D HAVE 0 = 3.
89. 4:46NOW, IF YOU RECALL FROM THE PREVIOUS VIDEO,
90. 4:48ALGEBRAICALLY IF THIS STATEMENT IS FALSE, WHICH IT IS,
91. 4:51THAT'S THE INDICATION THAT THIS SYSTEM HAS NO SOLUTION
92. 4:55WHICH ALSO MEANS THIS SYSTEM IS INCONSISTENT AND INDEPENDENT.
93. 5:01SO YOU CAN SEE THE ELIMINATION IS A NICE METHOD.
94. 5:03IT'S MUCH FASTER THAN TRYING TO GRAPH THESE LINEAR EQUATIONS.
95. 5:07LET'S GO AHEAD AND TAKE A LOOK AT AN APPLICATION.
96. 5:10SUNSET PHONE OFFERS 2 LONG DISTANCE PLANS.
97. 5:13PLAN A IS $5 PER MONTH AND 8 CENTS A MINUTE.
98. 5:18PLAN B HAS NO MONTHLY FEE BUT CHARGES 12 CENTS A MINUTE.
99. 5:22FOR WHAT NUMBER OF MINUTES WILL THE TWO PLANS COST THE SAME?
100. 5:27SO LET'S LET X = THE NUMBER OF MINUTES.
101. 5:32SO IF WE LET C = THE TOTAL COST FOR THE MONTH FOR PLAN A
102. 5:38WE HAVE A FIXED COST OF $5
103. 5:40AND A VARIABLE COST OF 8 CENTS PER MINUTE.
104. 5:44SO TO FIND THE TOTAL COST FOR THE MONTH
105. 5:46WE'D HAVE TO TAKE THE COST PER MINUTE,
106. 5:49MULTIPLY IT BY THE NUMBER OF MINUTES WHICH IS X
107. 5:52AND THEN ADD THE FIXED COST OF $5.
108. 5:56FOR PLAN B THE TOTAL COST FOR THE MONTH
109. 6:00IS A LITTLE BIT SIMPLER.
110. 6:01IT'S JUST 12 CENTS PER MINUTE
111. 6:04WHICH WOULD REPRESENT AS 0.12 x X.
112. 6:08AGAIN, THERE IS NO FIXED COSTS FOR PLAN B.
113. 6:11NOW THE QUESTION IS FOR WHAT NUMBER OF MINUTES
114. 6:13WILL THESE TWO PLANS BE THE SAME?
115. 6:15SO WE WANT TO KNOW WHEN THESE TWO C'S WILL BE EQUAL,
116. 6:19AND THAT WILL OCCUR WHEN 0.08X + 5 IS EQUAL TO 0.12X.
117. 6:26SO WHAT WE'RE REALLY DOING IS WE'RE PERFORMING SUBSTITUTION.
118. 6:29WE'LL TAKE THIS EXPRESSION AND SUB IT IN FOR THIS C
119. 6:34OR VICE VERSA.
120. 6:35THE RESULT WOULD BE 0.08X + 5.00 = 0.12X.
121. 6:45NOW IF WE DON'T LIKE TO WORK WITH DECIMALS
122. 6:47WE CAN MULTIPLY THIS ENTIRE EQUATION BY 100.
123. 6:50REMEMBER, MULTIPLYING BY 100
124. 6:52MOVES THE DECIMAL TO THE RIGHT 2 PLACES,
125. 6:55SO WE COULD SOLVE THE EQUATION 8X + 500 = 12X.
126. 7:01THEN WE WOULD SUBTRACT 8X ON BOTH SIDES TO GIVE US 4X.
127. 7:07DIVIDE BY 4.
128. 7:10X = 125.
129. 7:13THEREFORE THESE PLANS WOULD COST THE SAME
130. 7:17WHEN 125 MINUTES ARE USED FOR THE SAME MONTH.
131. 7:23I HOPE THAT HELPS EXPLAIN HOW TO USE THE ELIMINATION METHOD.
132. 7:27THANK YOU FOR WATCHING.