Solving Systems of Equations using Substitution — Transcript

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1. 0:01- WELCOME TO A VIDEO ON THE SUBSTITUTION METHOD
2. 0:03FOR SOLVING SYSTEMS OF EQUATIONS.
3. 0:08TO SOLVE A SYSTEM USING SUBSTITUTION,
4. 0:11FIRST WE'LL SOLVE FOR A VARIABLE IN EITHER ONE OF THE EQUATIONS
5. 0:15IF NEITHER EQUATION ALREADY HAS A VARIABLE ISOLATED.
6. 0:20STEP TWO, WE'LL SUBSTITUTE IN THE OTHER EQUATION
7. 0:22FOR THE VARIABLE ISOLATED.
8. 0:25STEP THREE, THE SUBSTITUTION
9. 0:28SHOULD RESULT IN AN EQUATION WITH ONE VARIABLE.
10. 0:31THEN WE'LL SOLVE THIS EQUATION.
11. 0:34STEP FOUR, WE'LL SUBSTITUTE THE SOLUTION OBTAINED
12. 0:36IN THE PREVIOUS STEP INTO ONE OF THE ORIGINAL EQUATIONS
13. 0:40TO SOLVE FOR THE OTHER VARIABLE.
14. 0:41AND THEN IT'S RECOMMENDED THAT WE CHECK OUR SOLUTION.
15. 0:46LET'S TAKE A LOOK AT THIS EXAMPLE
16. 0:47TO KIND OF SEE WHAT WE'RE TALKING ABOUT.
17. 0:50HERE WE SEE FROM THE FIRST EQUATION, Y = 3X.
18. 0:55WHAT THAT MEANS IS,
19. 0:56WHEREVER WE SEE Y IN THE SECOND EQUATION,
20. 0:59WE'LL REPLACE IT WITH 3X.
21. 1:03AGAIN SINCE Y = 3X HERE,
22. 1:06WE REPLACE THE Y WITH 3X IN THE SECOND EQUATION.
23. 1:10NOW LET'S REWRITE THE SECOND EQUATION WITH THE SUBSTITUTION.
24. 1:15THAT WOULD RESULT IN 2X.
25. 1:17AGAIN INSTEAD OF Y, WE'LL REPLACE IT
26. 1:19WITH 3X EQUAL TO -10.
27. 1:24COMBINE OUR LIKE TERMS.
28. 1:27DIVIDE BY 5.
29. 1:31X = -2.
30. 1:33AGAIN BECAUSE OF THE SUBSTITUTION,
31. 1:35WE OBTAINED AN EQUATION WITH ONE VARIABLE,
32. 1:38WHICH WE WERE THEN ABLE TO SOLVE.
33. 1:40BUT REMEMBER ONE SOLUTION CONSISTS
34. 1:43OF AN X-COORDINATE AND A Y-COORDINATE.
35. 1:46SO WHAT WE DO NOW IS WE TAKE OUR X-VALUE OF -2
36. 1:52AND SUBSTITUTE THIS BACK INTO EITHER EQUATION.
37. 1:55LET'S GO AHEAD AND PUT IT IN THE FIRST EQUATION.
38. 1:58FIRST EQUATION STATES Y = 3X.
39. 2:02SO IF WE NOW KNOW X = -2,
40. 2:06WE CAN SEE THAT THE Y-COORDINATE
41. 2:07OR THE Y-VALUE WOULD BE -6.
42. 2:10THEREFORE OUR SOLUTION IS X = -2, Y = -6.
43. 2:17NOW RELATE THIS TO OUR PREVIOUS VIDEO,
44. 2:19SOLVING SYSTEMS BY GRAPHING.
45. 2:21WHAT WE CAN DO IS GRAPH THESE TWO LINES
46. 2:24AND SEE IF THIS POINT IS THE POINT OF INTERSECTION.
47. 2:26REMEMBER, IN ORDER TO DO THAT, THOUGH,
48. 2:29WE DO HAVE TO SOLVE BOTH EQUATIONS FOR Y.
49. 2:31WELL, THE FIRST EQUATION IS ALREADY IN TERMS OF Y.
50. 2:37FOR THE SECOND EQUATION,
51. 2:38WE WOULD HAVE TO SUBTRACT 2X ON BOTH SIDES.
52. 2:44NOW WE CAN TYPE THESE INTO THE GRAPHING CALCULATOR.
53. 2:50NOW YOU NOTICE I'VE ALREADY DONE THAT
54. 2:51TO SAVE SOME TIME.
55. 2:52LET'S GO AHEAD AND HIT GRAPH.
56. 2:55THERE'S OUR POINT OF INTERSECTION.
57. 2:56IF WE HIT SECOND TRACE, IT BRINGS UP THE CALCULATION MENU.
58. 3:01OPTION 5, THEN HIT ENTER THREE TIMES.
59. 3:08WE CAN SEE THAT OUR SOLUTION IS CORRECT.
60. 3:14LET'S GO AHEAD AND TRY ANOTHER.
61. 3:17STEP ONE IS TO SOLVE ONE OF THE EQUATIONS
62. 3:19FOR ONE OF THE VARIABLES.
63. 3:21NOW IT WOULD BE EASY TO SOLVE THIS FIRST EQUATION FOR X.
64. 3:25LET'S GO AHEAD AND DO THAT.
65. 3:27IF WE ADD 2Y TO BOTH SIDES,
66. 3:29WE WOULD OBTAIN X = 1 + 2Y.
67. 3:34SO WHEREVER WE SEE X IN THE SECOND EQUATION,
68. 3:38WE'LL REPLACE IT WITH 1 + 2Y.
69. 3:41SO FOR THIS X, WE'LL SUBSTITUTE IN 1 + 2Y.
70. 3:51NOTICE WE ONLY HAVE ONE VARIABLE,
71. 3:53SO LET'S DISTRIBUTE AND SOLVE FOR Y.
72. 3:57NOW WHAT HAPPENS HERE, WE HAVE 6Y - 6Y.
73. 4:00THAT WOULD RESULT IN A ZERO.
74. 4:03SO WE'RE LEFT WITH 3 IS EQUAL TO -18.
75. 4:08REMEMBER WHEN WE SOLVE A SYSTEM OF EQUATIONS,
76. 4:10THERE'S THREE POSSIBILITIES:
77. 4:12ONE SOLUTION, NO SOLUTION,
78. 4:14OR AN INFINITE NUMBER OF SOLUTIONS.
79. 4:16SO ALGEBRAICALLY,
80. 4:17IF THE VARIABLES SIMPLIFY TO ZERO,
81. 4:20BUT THE RESULTING STATEMENT IS FALSE,
82. 4:24THIS IS A SIGN THAT THERE IS NO SOLUTION TO THIS SYSTEM.
83. 4:29WE COULD ALSO SAY
84. 4:30THE SYSTEM IS INCONSISTENT AND DEPENDENT.
85. 4:34GRAPHICALLY WHAT THAT MEANS IS, IF WE GRAPH THESE TWO LINES,
86. 4:37THE LINES SHOULD BE PARALLEL.
87. 4:41LET'S TAKE A LOOK AT THAT.
88. 4:42TO SAVE SOME TIME,
89. 4:43I'VE ALREADY SOLVED THESE TWO EQUATIONS FOR Y.
90. 4:46LET'S GO BACK TO THE GRAPHING CALCULATOR AND TYPE THESE IN.
91. 4:49REMEMBER WHEN YOU TYPE IN A FRACTIONAL SLOPE,
92. 4:53IT'S IMPORTANT TO INCLUDE THAT IN A SET OF PARENTHESES.
93. 5:02AND IF WE GRAPH THESE TWO LINES,
94. 5:04WHAT WE'LL NOTICE IS AS EXPECTED.
95. 5:07SINCE THE SLOPES ARE THE SAME
96. 5:09AND THE Y-INTERCEPTS ARE DIFFERENT,
97. 5:11THEY ARE PARALLEL LINES,
98. 5:12WHICH DOES VERIFY OUR ANSWER OF NO SOLUTION.
99. 5:19LET'S GO AHEAD AND TAKE A LOOK AT A COUPLE MORE.
100. 5:22SO WE FIRST HAVE TO SOLVE ONE OF THE EQUATIONS
101. 5:24FOR ONE OF THE VARIABLES.
102. 5:26AGAIN WE CAN SEE THAT IT WOULD BE EASIER
103. 5:28TO SOLVE THIS FIRST EQUATION FOR X,
104. 5:31SO LET'S GO AHEAD AND DO THAT BY ADDING 2Y TO BOTH SIDES.
105. 5:36SO SINCE X = 7 + 2Y,
106. 5:40WE CAN REPLACE THIS X WITH THAT EXPRESSION,
107. 5:44WHICH WOULD RESULT IN AN EQUATION
108. 5:45WITH JUST Ys AND THAT'S THE WHOLE POINT.
109. 5:49TO OBTAIN AN EQUATION WITH ONE VARIABLE.
110. 5:53NOW BE CAREFUL HERE.
111. 5:54WE HAVE TO THINK OF DISTRIBUTING A -3.
112. 6:01COMBINE OUR LIKE TERMS.
113. 6:03ADD 21 TO BOTH SIDES.
114. 6:07WELL -1 + 21 WOULD BE +20.
115. 6:12DIVIDE BY -4.
116. 6:15WE HAVE A Y-VALUE OF -5.
117. 6:18AGAIN WE DON'T WANT TO RUSH THROUGH THIS.
118. 6:20REMEMBER ONE SOLUTION CONSISTS OF AN X AND Y-VALUE.
119. 6:23SO FAR WE'VE ONLY FOUND THE Y-VALUE.
120. 6:25SO IN ORDER TO FIND THE X-VALUE,
121. 6:29WE NEED TO REPLACE Y WITH -5
122. 6:33IN ONE OF THESE TWO EQUATIONS OR A FORM OF IT.
123. 6:36HERE WE CAN SEE THAT THESE TWO EQUATIONS ARE EQUIVALENT.
124. 6:39SO TO FIND X, WE CAN JUST REPLACE WITH -5.
125. 6:48SO WE HAVE AN X-VALUE OF -3.
126. 6:55NOW TO VERIFY THIS SOLUTION,
127. 6:57WE COULD SUBSTITUTE THESE TWO VALUES INTO BOTH EQUATIONS.
128. 7:02I'D RECOMMEND THAT YOU WOULD DO THAT.
129. 7:03I'M NOT GOING TO DO THAT NOW JUST FOR THE SAKE OF TIME.
130. 7:06I WANT TO LOOK AT ONE MORE EXAMPLE.
131. 7:09AGAIN STEP ONE IS TO SOLVE ONE EQUATION FOR ONE VARIABLE.
132. 7:13I'M GOING TO WORK AGAIN WITH THIS FIRST EQUATION.
133. 7:16I'M GOING TO SOLVE THIS FOR Y.
134. 7:19SO THE FIRST STEP WOULD BE TO SUBTRACT 2X ON BOTH SIDES.
135. 7:22NOW BE CAREFUL HERE.
136. 7:23THERE'S A -Y,
137. 7:24SO WE'RE GOING TO HAVE A -Y IS EQUAL TO -5 - 2X.
138. 7:30WE WANT +Y, SO WHAT WE'RE GOING TO DO HERE
139. 7:33IS DIVIDE EVERYTHING BY -1.
140. 7:36OUR FINAL RESULTS WILL BE Y = +5 + 2X.
141. 7:44SO NOW WE'LL GO TO THE SECOND EQUATION,
142. 7:45AND WHERE WE SEE Y, WE'LL REPLACE IT WITH 5 + 2X.
143. 7:50SO THERE'S OUR Y. SO WHEN WE REWRITE THIS,
144. 7:53WE'RE GOING TO HAVE 4X - 2 x OUR Y,
145. 7:58WHICH IS 5 + 2X = -10.
146. 8:02AGAIN THE MAIN IDEA HERE IS WE'VE DONE SUBSTITUTION
147. 8:05TO FORM AN EQUATION WITH ONE UNKNOWN.
148. 8:08WE'LL DISTRIBUTE OUR -2 AND SOLVE FOR X.
149. 8:13NOW JUST LIKE A PREVIOUS EXAMPLE,
150. 8:15THE VARIABLE TERMS SIMPLIFY TO ZERO,
151. 8:17BUT NOW WE'RE LEFT WITH A TRUE STATEMENT.
152. 8:22SO WHEN THE VARIABLES SIMPLIFY TO ZERO,
153. 8:24BUT THE RESULT IS A TRUE STATEMENT,
154. 8:27THAT'S WHAT HAPPENS ALGEBRAICALLY.
155. 8:29WE HAVE AN INFINITE NUMBER OF SOLUTIONS.
156. 8:33AND IF YOU RECALL, IF WE WERE TO GRAPH THESE TWO LINES,
157. 8:35THEY WOULD ACTUALLY BE THE SAME LINE.
158. 8:38LET'S GO AHEAD AND SHOW THAT.
159. 8:40WE'VE ALREADY TAKEN THIS FIRST EQUATION AND SOLVED IT FOR Y,
160. 8:44BUT I'M GOING TO GO AHEAD AND REWRITE IN SLOPE-INTERCEPT FORM.
161. 8:47SO IT'D BE Y = 2X + 5.
162. 8:50NOW WE'LL TAKE EQUATION TWO AND SOLVE THIS FOR Y.
163. 8:55WE'LL SUBTRACT 4X ON BOTH SIDES.
164. 9:00THEN DIVIDE BY -2.
165. 9:04AND WE CAN SEE THAT OUR SECOND EQUATION
166. 9:06IS ALSO Y = 2X + 5.
167. 9:09SO WE WON'T TAKE THE TIME TO GRAPH THESE
168. 9:11BECAUSE WE CAN EASILY SEE THE SLOPES ARE THE SAME
169. 9:14AND THE Y-INTERCEPTS ARE THE SAME.
170. 9:16THEREFORE, WE WOULD HAVE THE SAME LINE.
171. 9:19AND IF YOUR INSTRUCTOR IS EMPHASIZING THE TERMINOLOGY,
172. 9:23WE COULD ALSO CLASSIFY THIS
173. 9:24AS AN INCONSISTENT, DEPENDENT SYSTEM.
174. 9:30THANK YOU FOR WATCHING.
175. 9:31AND I HOPE THAT HELPED.