1 00:00:00,000 --> 00:00:00,850 2 00:00:00,850 --> 00:00:03,560 We're on problem 27. 3 00:00:03,560 --> 00:00:06,780 And the question is, which equation best represents the 4 00:00:06,780 --> 00:00:08,210 graph above? 5 00:00:08,210 --> 00:00:10,420 So before even looking at the choices, let's see what we can 6 00:00:10,419 --> 00:00:11,390 figure out about the graph. 7 00:00:11,390 --> 00:00:12,830 So what's its y-intercept? 8 00:00:12,830 --> 00:00:16,070 So if we said that this is the equation of a line, we said y 9 00:00:16,070 --> 00:00:23,339 is equal to mx plus b where m is the slope and y is-- and b 10 00:00:23,339 --> 00:00:24,109 is the y-intercept. 11 00:00:24,109 --> 00:00:26,780 It's late, I'm getting slightly muddled. 12 00:00:26,780 --> 00:00:27,780 So what's the y-intercept? 13 00:00:27,780 --> 00:00:30,530 Well when x is equal to 0, y is equal to 0. 14 00:00:30,530 --> 00:00:31,990 So this is going to be equal to 0. 15 00:00:31,989 --> 00:00:33,509 The y-intercept is 0. 16 00:00:33,509 --> 00:00:36,309 When x is equal to 0, why is equal to 0. 17 00:00:36,310 --> 00:00:38,190 So the y intercept is 0. 18 00:00:38,189 --> 00:00:41,579 So we know that this has a form y is equal to mx where m 19 00:00:41,579 --> 00:00:42,839 is a slope. 20 00:00:42,840 --> 00:00:43,810 Let's figure out the slope. 21 00:00:43,810 --> 00:00:48,429 Slope is equal to change in y for a given change in x or 22 00:00:48,429 --> 00:00:50,015 change in y over change in x. 23 00:00:50,015 --> 00:00:54,929 So when we increase x by 1, how much are we increase-- or 24 00:00:54,929 --> 00:00:55,969 decrease y by? 25 00:00:55,969 --> 00:01:00,269 Well then y increases by 2. 26 00:01:00,270 --> 00:01:04,650 So we could say that equals y changes by 2, by plus 2 when x 27 00:01:04,650 --> 00:01:06,609 changes by plus 1. 28 00:01:06,609 --> 00:01:09,429 So we get the slope is equal to 2, so the equation of this 29 00:01:09,430 --> 00:01:11,740 line is y is equal to 2x. 30 00:01:11,739 --> 00:01:13,839 Which is choice B. 31 00:01:13,840 --> 00:01:19,040 32 00:01:19,040 --> 00:01:20,240 Next problem. 33 00:01:20,239 --> 00:01:23,689 Which point lies on the line defined by 3x plus 34 00:01:23,689 --> 00:01:25,469 6y is equal to 2? 35 00:01:25,469 --> 00:01:27,409 Well the best thing to do is just probably to substitute 36 00:01:27,409 --> 00:01:29,989 these numbers in for x and y and see which one works. 37 00:01:29,989 --> 00:01:32,429 So here x is 0, y is 2. 38 00:01:32,430 --> 00:01:32,780 So let's see. 39 00:01:32,780 --> 00:01:40,659 3 times 0 plus 6 times 2 is equal to 0 plus 12. 40 00:01:40,659 --> 00:01:42,899 Doesn't equal 2, it equals 12. 41 00:01:42,900 --> 00:01:44,590 This one doesn't work. 42 00:01:44,590 --> 00:01:46,609 I'm just taking 3 times x plus 6 times y and 43 00:01:46,609 --> 00:01:47,750 seeing what it equals. 44 00:01:47,750 --> 00:01:53,079 In this case, we have 3 times 0 plus 6 times y. 45 00:01:53,079 --> 00:01:55,799 Plus 6 times 6. 46 00:01:55,799 --> 00:01:58,039 Well that's 0 plus 36. 47 00:01:58,040 --> 00:01:59,600 That doesn't equal 2. 48 00:01:59,599 --> 00:02:01,419 Can't be that choice. 49 00:02:01,420 --> 00:02:07,000 This one, we have 3, this 3, times 1. 50 00:02:07,000 --> 00:02:09,788 Plus 6 times this y. 51 00:02:09,788 --> 00:02:13,530 6 times minus 1/6. 52 00:02:13,530 --> 00:02:14,259 So let's see. 53 00:02:14,259 --> 00:02:16,039 That's 3. 54 00:02:16,039 --> 00:02:18,349 That's equal to 3. 55 00:02:18,349 --> 00:02:21,509 And then, 6 times 1/6 is 1, but we have a 56 00:02:21,509 --> 00:02:22,239 negative out there. 57 00:02:22,240 --> 00:02:23,340 So it's minus 1. 58 00:02:23,340 --> 00:02:25,170 That is equal to 2. 59 00:02:25,169 --> 00:02:26,209 So that works. 60 00:02:26,210 --> 00:02:30,689 3 times 1 plus 6 times negative 1/6 is equal to 2. 61 00:02:30,689 --> 00:02:34,189 So our answer is C. 62 00:02:34,189 --> 00:02:35,990 Problem 29. 63 00:02:35,990 --> 00:02:39,469 Let me see if I need to cut and paste this one in. 64 00:02:39,469 --> 00:02:40,830 All right. 65 00:02:40,830 --> 00:02:43,390 So yeah, I think it's a good idea. 66 00:02:43,389 --> 00:02:44,819 Let me see. 67 00:02:44,819 --> 00:02:45,900 Copy and paste it. 68 00:02:45,900 --> 00:02:49,689 Let me copy and paste the next several problems. We could 69 00:02:49,689 --> 00:02:53,819 just move-- be streamlined about it. 70 00:02:53,819 --> 00:02:58,069 71 00:02:58,069 --> 00:03:00,079 OK, they all fit. 72 00:03:00,080 --> 00:03:01,330 Good enough. 73 00:03:01,330 --> 00:03:03,350 74 00:03:03,349 --> 00:03:06,530 What is the equation of the line that has slope of 4 and 75 00:03:06,530 --> 00:03:09,770 passes through the point 3 comma minus 10? 76 00:03:09,770 --> 00:03:12,409 So if the slope is 4, we know that the line-- so just 77 00:03:12,409 --> 00:03:15,639 rewrite it in slope y-intercept form again. 78 00:03:15,639 --> 00:03:17,169 mx plus b. 79 00:03:17,169 --> 00:03:19,929 They're telling us that the slope is 4. 80 00:03:19,930 --> 00:03:25,260 So we know the equation of the line is y is equal to 4x plus 81 00:03:25,259 --> 00:03:26,759 the y-intercept. 82 00:03:26,759 --> 00:03:28,689 And then we could figure out the y-intercept by 83 00:03:28,689 --> 00:03:31,780 substituting in this point that they say it goes through. 84 00:03:31,780 --> 00:03:35,659 So it goes through the point 3 comma negative 10. 85 00:03:35,659 --> 00:03:41,949 So y is equal to minus 10 when x is equal to 3. 86 00:03:41,949 --> 00:03:44,019 So 4 times x. 87 00:03:44,020 --> 00:03:46,250 x is equal to 3. 88 00:03:46,250 --> 00:03:48,229 Plus b. 89 00:03:48,229 --> 00:03:49,039 So what is that? 90 00:03:49,039 --> 00:03:53,669 That's minus 10 is equal to 12 plus b. 91 00:03:53,669 --> 00:03:57,659 We could subtract 12 from both sides of this equation, and we 92 00:03:57,659 --> 00:03:59,979 get minus 22. 93 00:03:59,979 --> 00:04:03,259 Minus 10 minus 12 is minus 22. 94 00:04:03,259 --> 00:04:04,709 This 12 obviously goes away. 95 00:04:04,710 --> 00:04:05,830 Is equal to b. 96 00:04:05,830 --> 00:04:11,920 So the equation of our line is y is equal to 4x plus b, which 97 00:04:11,919 --> 00:04:15,119 we just figured out is minus 22. 98 00:04:15,120 --> 00:04:17,019 4x minus 22. 99 00:04:17,019 --> 00:04:19,060 That's choice A. 100 00:04:19,060 --> 00:04:21,838 101 00:04:21,838 --> 00:04:25,379 Problem 30. 102 00:04:25,379 --> 00:04:28,269 The date in the table show the cost of renting a bicycle by 103 00:04:28,269 --> 00:04:29,799 the hour, including a deposit. 104 00:04:29,800 --> 00:04:32,430 105 00:04:32,430 --> 00:04:36,090 If hours, h, were graphed on the horizontal axis-- let me 106 00:04:36,089 --> 00:04:38,379 see if I can draw that. 107 00:04:38,379 --> 00:04:45,009 So that is on the horizontal axis I have hours, h. 108 00:04:45,009 --> 00:04:46,159 So this is h. 109 00:04:46,160 --> 00:04:48,950 And costs were graphed on the vertical axis, so let me draw 110 00:04:48,949 --> 00:04:50,199 the vertical axis. 111 00:04:50,199 --> 00:04:52,879 112 00:04:52,879 --> 00:04:57,029 So cost is on the vertical axis. 113 00:04:57,029 --> 00:05:00,389 What would be the equation of the line that fits the data? 114 00:05:00,389 --> 00:05:03,969 OK, so they just want to know cost as a function of hours. 115 00:05:03,970 --> 00:05:06,160 So let's see. 116 00:05:06,160 --> 00:05:07,210 It's a linear relationship. 117 00:05:07,209 --> 00:05:10,169 We could treat this just like any other line. 118 00:05:10,170 --> 00:05:13,100 So if we write it as y equals mx plus b. 119 00:05:13,100 --> 00:05:14,060 I didn't even have to do this. 120 00:05:14,060 --> 00:05:18,300 So y is equal to mx plus b. 121 00:05:18,300 --> 00:05:23,449 So m, or the slope, is equal to change in y over-- or let 122 00:05:23,449 --> 00:05:24,889 me write it differently. 123 00:05:24,889 --> 00:05:28,129 Because we want it in terms of-- so we could write c is 124 00:05:28,129 --> 00:05:32,939 equal to the slope times the hours plus b. 125 00:05:32,939 --> 00:05:36,579 Instead of x, we have hours, and instead of y, we have the 126 00:05:36,579 --> 00:05:38,729 cost. 127 00:05:38,730 --> 00:05:40,210 So what's the slope? 128 00:05:40,209 --> 00:05:44,469 Slope is equal to change in the dependent variable, change 129 00:05:44,470 --> 00:05:47,840 in cost, divided by change in the independent variable, 130 00:05:47,839 --> 00:05:49,931 divided by change in hours. 131 00:05:49,932 --> 00:05:52,410 So let's see. 132 00:05:52,410 --> 00:05:57,350 If when our hours increased by 3, how much did our 133 00:05:57,350 --> 00:05:58,450 change in c go by? 134 00:05:58,449 --> 00:06:03,389 When we go from 2 to 5, so if we say 5 minus 2, that's how 135 00:06:03,389 --> 00:06:05,029 much we changed in hours. 136 00:06:05,029 --> 00:06:06,609 That's delta h. 137 00:06:06,610 --> 00:06:09,100 Then how much did we change in cost? 138 00:06:09,100 --> 00:06:12,590 Well it's 30 minus 15. 139 00:06:12,589 --> 00:06:15,859 So it equals 15 over 3. 140 00:06:15,860 --> 00:06:17,780 which is equal to 5. 141 00:06:17,779 --> 00:06:19,389 So we figured out the slope. 142 00:06:19,389 --> 00:06:21,329 Slope is equal to 5. 143 00:06:21,329 --> 00:06:25,129 Now we just have to figure out the y-intercept. 144 00:06:25,129 --> 00:06:28,709 So this equation, we could rewrite it as cost is equal to 145 00:06:28,709 --> 00:06:33,699 the slope, which is 5, times the hours plus whatever our 146 00:06:33,699 --> 00:06:34,889 y-intercept is. 147 00:06:34,889 --> 00:06:36,629 And then we just have to substitute one of the points 148 00:06:36,629 --> 00:06:40,459 in to solve for b, just like we did in the previous video. 149 00:06:40,459 --> 00:06:44,569 So let's substitute the point when h is equal to 2, c is 150 00:06:44,569 --> 00:06:45,409 equal to 15. 151 00:06:45,410 --> 00:06:48,220 So the point 2 comma 15. 152 00:06:48,220 --> 00:06:51,710 So when h is equal to 2, c is equal to 15. 153 00:06:51,709 --> 00:06:56,259 c is equal to 15 when h is equal to 2. 154 00:06:56,259 --> 00:06:58,039 And now we can just solve for b. 155 00:06:58,040 --> 00:07:02,000 So you get 15 is equal to 10 plus b. 156 00:07:02,000 --> 00:07:08,050 Subtract 10 from both sides, you get b is equal to what? 157 00:07:08,050 --> 00:07:09,370 5. 158 00:07:09,370 --> 00:07:16,649 So the equation of this line is cost is equal to 5 times h 159 00:07:16,649 --> 00:07:20,859 plus our y-intercept, or plus our c-intercept we could say. 160 00:07:20,860 --> 00:07:23,379 So plus 5. 161 00:07:23,379 --> 00:07:27,990 5h plus 5, and that is choice C. 162 00:07:27,990 --> 00:07:29,199 And if you try it out with these points, 163 00:07:29,199 --> 00:07:30,939 it should all work. 164 00:07:30,939 --> 00:07:32,305 OK, next problem. 165 00:07:32,305 --> 00:07:35,240 166 00:07:35,240 --> 00:07:37,410 Let me see. 167 00:07:37,410 --> 00:07:41,021 I think this cutting and pasting works out well. 168 00:07:41,021 --> 00:07:43,220 I think I have time for at least two more. 169 00:07:43,220 --> 00:07:45,980 170 00:07:45,980 --> 00:07:47,759 OK, let me clear this. 171 00:07:47,759 --> 00:07:50,819 172 00:07:50,819 --> 00:07:53,719 Clear this one right here. 173 00:07:53,720 --> 00:07:56,820 Let me paste it in. 174 00:07:56,819 --> 00:08:01,500 OK, it says, some ordered pairs for a linear function of 175 00:08:01,500 --> 00:08:03,790 x are given in the table below. 176 00:08:03,790 --> 00:08:05,270 It's kind of like the last one. 177 00:08:05,269 --> 00:08:07,399 Which of the following equations was used to generate 178 00:08:07,399 --> 00:08:08,849 the table above? 179 00:08:08,850 --> 00:08:11,070 Well we could do the same thing we did the last time. 180 00:08:11,069 --> 00:08:13,500 We could say OK, this is going to have some relation. 181 00:08:13,500 --> 00:08:16,750 You know y is equal to mx plus b. 182 00:08:16,750 --> 00:08:18,350 It's going to be a line and this is the 183 00:08:18,350 --> 00:08:20,180 slope y-intercept form. 184 00:08:20,180 --> 00:08:21,670 And they tell us it's the linear function. 185 00:08:21,670 --> 00:08:23,280 So that's why we can immediately say that's going 186 00:08:23,279 --> 00:08:23,829 to be a line. 187 00:08:23,829 --> 00:08:26,909 There's no x squared or anything like that. 188 00:08:26,910 --> 00:08:30,980 So a good place to start is just with the slope. 189 00:08:30,980 --> 00:08:37,570 So slope is equal to change in y over change in x. 190 00:08:37,570 --> 00:08:37,970 So let's see. 191 00:08:37,970 --> 00:08:40,899 When x goes from 1 to 3, what does y go for. 192 00:08:40,899 --> 00:08:44,090 It goes from 1 to 7. 193 00:08:44,090 --> 00:08:48,830 So change in x is equal to 3 minus 1 or just 2. 194 00:08:48,830 --> 00:08:50,980 And what's change in y? 195 00:08:50,980 --> 00:08:54,230 7 minus 1. 196 00:08:54,230 --> 00:08:58,680 So that is equal to 6 over-- oh, no. 197 00:08:58,679 --> 00:08:58,899 Sorry. 198 00:08:58,899 --> 00:09:01,559 Change in x was 3 minus 1. 199 00:09:01,559 --> 00:09:03,139 Sorry about that. 200 00:09:03,139 --> 00:09:04,019 3 minus 1. 201 00:09:04,019 --> 00:09:08,289 So it's 6 over 2. 202 00:09:08,289 --> 00:09:13,939 When x changes by 2-- that's this 2, change in x-- y 203 00:09:13,940 --> 00:09:15,090 changes by 6. 204 00:09:15,090 --> 00:09:16,920 That's change in y. 205 00:09:16,919 --> 00:09:19,259 So the slope is 3. 206 00:09:19,259 --> 00:09:23,659 So the equation of our line is going to be y is equal to 3x 207 00:09:23,659 --> 00:09:25,899 plus some y-intercept. 208 00:09:25,899 --> 00:09:27,659 And now we could substitute some points in. 209 00:09:27,659 --> 00:09:28,490 Let's substitute this one. 210 00:09:28,490 --> 00:09:29,669 This 1, 1 seems easy. 211 00:09:29,669 --> 00:09:33,529 So y is equal to 1 when x is equal to 1. 212 00:09:33,529 --> 00:09:36,500 213 00:09:36,500 --> 00:09:40,009 So we get 1 is equal to 3 plus b. 214 00:09:40,009 --> 00:09:44,929 Subtract 3 from both sides of this equation and we get what? 215 00:09:44,929 --> 00:09:48,179 1 minus 3 is minus 2 is equal to-- obviously, this 216 00:09:48,179 --> 00:09:50,279 disappears, 3 minus 3. 217 00:09:50,279 --> 00:09:52,449 So b is equal to minus 2. 218 00:09:52,450 --> 00:09:54,759 So the equation of our line, we have a slope of 3 and a 219 00:09:54,759 --> 00:09:56,080 y-intercept of minus 2. 220 00:09:56,080 --> 00:10:01,840 The equation is going to be y is equal to 3x plus b. 221 00:10:01,840 --> 00:10:02,960 Well b is minus 2. 222 00:10:02,960 --> 00:10:05,129 So minus 2. 223 00:10:05,129 --> 00:10:06,939 And that is choice C. 224 00:10:06,940 --> 00:10:13,310 225 00:10:13,309 --> 00:10:14,729 32. 226 00:10:14,730 --> 00:10:18,870 The equation of line l is 6x plus 5y equals 3. 227 00:10:18,870 --> 00:10:24,570 And equation of line q is 5x minus 6y is equal to 0. 228 00:10:24,570 --> 00:10:27,070 Which of the statements about the two lines are true? 229 00:10:27,070 --> 00:10:27,820 OK, let's see. 230 00:10:27,820 --> 00:10:29,360 They have the same y intercept. 231 00:10:29,360 --> 00:10:30,220 They're parallel. 232 00:10:30,220 --> 00:10:33,050 They have the same x-intercept. 233 00:10:33,049 --> 00:10:36,509 Lines l and q are perpendicular. 234 00:10:36,509 --> 00:10:38,970 Well I'll tell you how you can identify perpendicular lines. 235 00:10:38,970 --> 00:10:41,470 They have the negative inverse of each other's slope. 236 00:10:41,470 --> 00:10:43,710 But I don't know if that's going to come into play yet. 237 00:10:43,710 --> 00:10:47,960 I always find it easier to look at these lines in kind of 238 00:10:47,960 --> 00:10:49,310 the slope y-intercept form. 239 00:10:49,309 --> 00:10:49,939 So let's do that. 240 00:10:49,940 --> 00:10:57,540 So line l can be 6x plus 5y is equal to 3. 241 00:10:57,539 --> 00:11:01,159 So if we were to subtract 6x from both sides you get 5y is 242 00:11:01,159 --> 00:11:04,559 equal to minus 6x plus 3. 243 00:11:04,559 --> 00:11:08,299 And then you divide both sides by 5 and you get y is equal to 244 00:11:08,299 --> 00:11:14,159 minus 6/5 x plus 3/5. 245 00:11:14,159 --> 00:11:18,379 So this is line l right here. 246 00:11:18,379 --> 00:11:20,100 I'll do line q in a different color. 247 00:11:20,100 --> 00:11:27,180 So q is 5x minus 6y is equal to 0. 248 00:11:27,179 --> 00:11:29,579 So let's subtract 5x from both sides. 249 00:11:29,580 --> 00:11:35,080 You get minus 6y is equal to minus 5x. 250 00:11:35,080 --> 00:11:39,759 If you divide both sides by negative 6, you get y is equal 251 00:11:39,759 --> 00:11:42,000 to-- minus 5 divided by negative 6. 252 00:11:42,000 --> 00:11:46,009 That's 5/6 x. 253 00:11:46,009 --> 00:11:48,470 Well, it actually looks like D comes into play. 254 00:11:48,470 --> 00:11:52,060 Because the slope of this one is minus 6/5. 255 00:11:52,059 --> 00:11:54,109 That's the slope of line l. 256 00:11:54,110 --> 00:11:59,620 The slope of line q is the inverse of that made negative. 257 00:11:59,620 --> 00:12:01,389 At this point it's a good thing to memorize. 258 00:12:01,389 --> 00:12:04,580 I haven't proven it to you yet. 259 00:12:04,580 --> 00:12:07,670 If you want a line that has a slope perpendicular to this 260 00:12:07,669 --> 00:12:10,589 one, or a line that is perpendicular to this one will 261 00:12:10,590 --> 00:12:12,470 have a negative inverse slope. 262 00:12:12,470 --> 00:12:16,629 So the inverse of this is minus 5/6 and the negative 263 00:12:16,629 --> 00:12:19,029 inverse is plus 5/6. 264 00:12:19,029 --> 00:12:21,649 So this is the negative inverse of that. 265 00:12:21,649 --> 00:12:24,519 So these lines are actually perpendicular. 266 00:12:24,519 --> 00:12:27,740 So that is D. 267 00:12:27,740 --> 00:12:29,560 See you in the next video.