1 00:00:00,000 --> 00:00:00,700 2 00:00:00,700 --> 00:00:06,250 A line passes through the points negative 3, 6 and 6, 0. 3 00:00:06,250 --> 00:00:09,390 Find the equation of this line in point slope form, slope 4 00:00:09,390 --> 00:00:11,860 intercept form, standard form. 5 00:00:11,859 --> 00:00:13,969 And the way to think about these, these are just three 6 00:00:13,970 --> 00:00:16,030 different ways of writing the same equation. 7 00:00:16,030 --> 00:00:18,220 So if you give me one of them, we can manipulate it to get 8 00:00:18,219 --> 00:00:19,230 any of the other ones. 9 00:00:19,230 --> 00:00:23,890 But just so you know what these are, point slope form, 10 00:00:23,890 --> 00:00:28,500 let's say the point x1, y1 are, let's say that that is a 11 00:00:28,500 --> 00:00:30,160 point on the line. 12 00:00:30,160 --> 00:00:32,899 And when someone puts this little subscript here, so if 13 00:00:32,899 --> 00:00:34,750 they just write an x, that means we're talking about a 14 00:00:34,750 --> 00:00:36,689 variable that can take on any value. 15 00:00:36,689 --> 00:00:39,769 If someone writes x with a subscript 1 and a y with a 16 00:00:39,770 --> 00:00:45,150 subscript 1, that's like saying a particular value x 17 00:00:45,149 --> 00:00:48,449 and a particular value of y, or a particular coordinate. 18 00:00:48,450 --> 00:00:50,030 And you'll see that when we do the example. 19 00:00:50,030 --> 00:00:52,480 But point slope form says that, look, if I know a 20 00:00:52,479 --> 00:00:56,019 particular point, and if I know the slope of the line, 21 00:00:56,020 --> 00:00:58,750 then putting that line in point slope form would be y 22 00:00:58,750 --> 00:01:05,209 minus y1 is equal to m times x minus x1. 23 00:01:05,209 --> 00:01:08,829 So, for example, and we'll do that in this video, if the 24 00:01:08,829 --> 00:01:13,170 point negative 3 comma 6 is on the line, then we'd say y 25 00:01:13,170 --> 00:01:19,510 minus 6 is equal to m times x minus negative 3, so it'll end 26 00:01:19,510 --> 00:01:21,120 up becoming x plus 3. 27 00:01:21,120 --> 00:01:23,115 So this is a particular x, and a particular y. 28 00:01:23,114 --> 00:01:25,129 It could be a negative 3 and 6. 29 00:01:25,129 --> 00:01:26,869 So that's point slope form. 30 00:01:26,870 --> 00:01:32,790 Slope intercept form is y is equal to mx plus b, where once 31 00:01:32,790 --> 00:01:36,810 again m is the slope, b is the y-intercept-- where does the 32 00:01:36,810 --> 00:01:40,180 line intersect the y-axis-- what value does y take 33 00:01:40,180 --> 00:01:41,680 on when x is 0? 34 00:01:41,680 --> 00:01:47,680 And then standard form is the form ax plus by is equal to c, 35 00:01:47,680 --> 00:01:50,810 where these are just two numbers, essentially. 36 00:01:50,810 --> 00:01:52,960 They really don't have any interpretation 37 00:01:52,959 --> 00:01:54,429 directly on the graph. 38 00:01:54,430 --> 00:01:57,710 So let's do this, let's figure out all of these forms. So the 39 00:01:57,709 --> 00:01:59,899 first thing we want to do is figure out the slope. 40 00:01:59,900 --> 00:02:02,609 Once we figure out the slope, then point slope form is 41 00:02:02,609 --> 00:02:05,819 actually very, very, very straightforward to calculate. 42 00:02:05,819 --> 00:02:12,120 So, just to remind ourselves, slope, which is equal to m, 43 00:02:12,120 --> 00:02:17,110 which is going to be equal to the change in y over the 44 00:02:17,110 --> 00:02:18,480 change in x. 45 00:02:18,479 --> 00:02:19,939 Now what is the change in y? 46 00:02:19,939 --> 00:02:23,609 If we view this as our end point, if we imagine that we 47 00:02:23,610 --> 00:02:26,190 are going from here to that point, what is 48 00:02:26,189 --> 00:02:27,460 the change in y? 49 00:02:27,460 --> 00:02:30,860 Well, we have our end point, which is 0, y ends up at the 50 00:02:30,860 --> 00:02:33,300 0, and y was at 6. 51 00:02:33,300 --> 00:02:38,650 So, our finishing y point is 0, our starting y point is 6. 52 00:02:38,650 --> 00:02:41,870 What was our finishing x point, or x-coordinate? 53 00:02:41,870 --> 00:02:44,810 Our finishing x-coordinate was 6. 54 00:02:44,810 --> 00:02:46,689 Let me make this very clear, I don't want to confuse you. 55 00:02:46,689 --> 00:02:53,719 So this 0, we have that 0, that is that 0 right there. 56 00:02:53,719 --> 00:02:59,330 And then we have this 6, which was our starting y point, that 57 00:02:59,330 --> 00:03:01,580 is that 6 right there. 58 00:03:01,580 --> 00:03:06,000 And then we want our finishing x value-- that is that 6 right 59 00:03:06,000 --> 00:03:10,050 there, or that 6 right there-- and we want to subtract from 60 00:03:10,050 --> 00:03:14,080 that our starting x value. 61 00:03:14,080 --> 00:03:16,800 Well, our starting x value is that right over there, that's 62 00:03:16,800 --> 00:03:19,910 that negative 3. 63 00:03:19,909 --> 00:03:21,879 And just to make sure we know what we're doing, this 64 00:03:21,879 --> 00:03:25,439 negative 3 is that negative 3, right there. 65 00:03:25,439 --> 00:03:29,500 I'm just saying, if we go from that point to that point, our 66 00:03:29,500 --> 00:03:31,639 y went down by 6, right? 67 00:03:31,639 --> 00:03:32,959 We went from 6 to 0. 68 00:03:32,960 --> 00:03:37,650 Our y went down by 6. 69 00:03:37,650 --> 00:03:40,420 So we get 0 minus 6 is negative 6. 70 00:03:40,419 --> 00:03:41,069 That makes sense. 71 00:03:41,069 --> 00:03:43,150 Y went down by 6. 72 00:03:43,150 --> 00:03:45,180 And, if we went from that point to that point, what 73 00:03:45,180 --> 00:03:46,060 happened to x? 74 00:03:46,060 --> 00:03:49,560 We went from negative 3 to 6, it should go up by 9. 75 00:03:49,560 --> 00:03:54,870 And if you calculate this, take your 6 minus negative 3, 76 00:03:54,870 --> 00:03:58,550 that's the same thing as 6 plus 3, that is 9. 77 00:03:58,550 --> 00:04:00,120 And what is negative 6/9? 78 00:04:00,120 --> 00:04:04,060 Well, if you simplify it, it is negative 2/3. 79 00:04:04,060 --> 00:04:06,949 You divide the numerator and the denominator by 3. 80 00:04:06,949 --> 00:04:09,859 So that is our slope, negative 2/3. 81 00:04:09,860 --> 00:04:12,800 So we're pretty much ready to use point slope form. 82 00:04:12,800 --> 00:04:15,760 We have a point, we could pick one of these points, I'll just 83 00:04:15,759 --> 00:04:18,149 go with the negative 3, 6. 84 00:04:18,149 --> 00:04:19,269 And we have our slope. 85 00:04:19,269 --> 00:04:20,910 So let's put it in point slope form. 86 00:04:20,910 --> 00:04:29,050 87 00:04:29,050 --> 00:04:32,910 All we have to do is we say y minus-- now we could have 88 00:04:32,910 --> 00:04:35,030 taken either of these points, I'll take this one-- so y 89 00:04:35,029 --> 00:04:42,989 minus the y value over here, so y minus 6 is equal to our 90 00:04:42,990 --> 00:04:51,480 slope, which is negative 2/3 times x minus our 91 00:04:51,480 --> 00:04:51,930 x-coordinate. 92 00:04:51,930 --> 00:04:55,490 Well, our x-coordinate, so x minus our x-coordinate is 93 00:04:55,490 --> 00:05:00,370 negative 3, x minus negative 3, and we're done. 94 00:05:00,370 --> 00:05:01,569 We can simplify it a little bit. 95 00:05:01,569 --> 00:05:08,649 This becomes y minus 6 is equal to negative 2/3 times x. 96 00:05:08,649 --> 00:05:12,609 x minus negative 3 is the same thing as x plus 3. 97 00:05:12,610 --> 00:05:15,759 This is our point slope form. 98 00:05:15,759 --> 00:05:19,269 Now, we can literally just algebraically manipulate this 99 00:05:19,269 --> 00:05:23,500 guy right here to put it into our slope intercept form. 100 00:05:23,500 --> 00:05:24,129 Let's do that. 101 00:05:24,129 --> 00:05:27,560 So let's do slope intercept in orange. 102 00:05:27,560 --> 00:05:30,959 So we have slope intercept. 103 00:05:30,959 --> 00:05:35,199 104 00:05:35,199 --> 00:05:37,039 So what can we do here to simplify this? 105 00:05:37,040 --> 00:05:41,510 Well, we can multiply out the negative 2/3, so you get y 106 00:05:41,509 --> 00:05:44,149 minus 6 is equal to-- I'm just distributing the negative 107 00:05:44,149 --> 00:05:48,949 2/3-- so negative 2/3 times x is negative 2/3 x. 108 00:05:48,949 --> 00:05:53,889 And then negative 2/3 times 3 is negative 2. 109 00:05:53,889 --> 00:05:56,149 And now to get it in slope intercept form, we just have 110 00:05:56,149 --> 00:05:59,269 to add the 6 to both sides so we get rid of it on the 111 00:05:59,269 --> 00:06:02,379 left-hand side, so let's add 6 to both 112 00:06:02,379 --> 00:06:04,399 sides of this equation. 113 00:06:04,399 --> 00:06:06,009 Left-hand side of the equation, we're just left with 114 00:06:06,009 --> 00:06:08,089 a y, these guys cancel out. 115 00:06:08,089 --> 00:06:12,829 You get a y is equal to negative 2/3 x. 116 00:06:12,829 --> 00:06:15,810 Negative 2 plus 6 is plus 4. 117 00:06:15,810 --> 00:06:19,889 So there you have it, that is our slope intercept form, mx 118 00:06:19,889 --> 00:06:22,589 plus b, that's our y-intercept. 119 00:06:22,589 --> 00:06:24,479 Now the last thing we need to do is get it into 120 00:06:24,480 --> 00:06:25,730 the standard form. 121 00:06:25,730 --> 00:06:30,439 122 00:06:30,439 --> 00:06:32,839 So once again, we just have to algebraically manipulate it so 123 00:06:32,839 --> 00:06:34,500 that the x's and the y's are both on 124 00:06:34,500 --> 00:06:36,129 this side of the equation. 125 00:06:36,129 --> 00:06:40,139 So let's just add 2/3 x to both sides of this equation. 126 00:06:40,139 --> 00:06:42,139 So I'll start it here. 127 00:06:42,139 --> 00:06:45,949 So we have y is equal to negative 2/3 x plus 4, that's 128 00:06:45,949 --> 00:06:47,539 slope intercept form. 129 00:06:47,540 --> 00:06:53,790 Let's added 2/3 x, so plus 2/3 x to both 130 00:06:53,790 --> 00:06:55,040 sides of this equation. 131 00:06:55,040 --> 00:06:57,650 132 00:06:57,649 --> 00:07:00,060 I'm doing that so it I don't have this 2/3 x on the 133 00:07:00,060 --> 00:07:02,350 right-hand side, this negative 2/3 x. 134 00:07:02,350 --> 00:07:05,010 So the left-hand side of the equation-- I scrunched it up a 135 00:07:05,009 --> 00:07:07,730 little bit, maybe more than I should have-- the left-hand 136 00:07:07,730 --> 00:07:09,890 side of this equation is what? 137 00:07:09,889 --> 00:07:16,389 It is 2/3 x, because 2 over 3x, plus this y, that's my 138 00:07:16,389 --> 00:07:20,219 left-hand side, is equal to-- these guys cancel out-- is 139 00:07:20,220 --> 00:07:22,650 equal to 4. 140 00:07:22,649 --> 00:07:25,579 So this, by itself, we are in standard form, this is the 141 00:07:25,579 --> 00:07:26,789 standard form of the equation. 142 00:07:26,790 --> 00:07:30,030 If we want it to look, make it look extra clean and have no 143 00:07:30,029 --> 00:07:32,339 fractions here, we could multiply both sides of this 144 00:07:32,339 --> 00:07:33,859 equation by 3. 145 00:07:33,860 --> 00:07:36,400 If we do that, what do we get? 146 00:07:36,399 --> 00:07:40,219 2/3 x times 3 is just 2x. 147 00:07:40,220 --> 00:07:43,440 y times 3 is 3y. 148 00:07:43,439 --> 00:07:45,699 And then 4 times 3 is 12. 149 00:07:45,699 --> 00:07:47,620 These are the same equations, I just multiplied 150 00:07:47,620 --> 00:07:48,759 every term by 3. 151 00:07:48,759 --> 00:07:50,189 If you do it to the left-hand side, you can do to the 152 00:07:50,189 --> 00:07:52,680 right-hand side-- or you have to do to the right-hand side-- 153 00:07:52,680 --> 00:07:56,160 and we are in standard form.