[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:01.48,0:00:05.73,Default,,0000,0000,0000,,This unit is about the equation\Nof a straight line.
Dialogue: 0,0:00:06.57,0:00:10.04,Default,,0000,0000,0000,,The equation of a straight line\Ncan take different forms
Dialogue: 0,0:00:10.04,0:00:13.51,Default,,0000,0000,0000,,depending upon the information\Nthat we know about the line.
Dialogue: 0,0:00:13.51,0:00:16.98,Default,,0000,0000,0000,,Let's start by a specific\Nexample. Suppose we've got some.
Dialogue: 0,0:00:17.50,0:00:22.66,Default,,0000,0000,0000,,Points. Labeled by their X&Y\Ncoordinates. So suppose we have
Dialogue: 0,0:00:22.66,0:00:25.90,Default,,0000,0000,0000,,a point where X is not why is 2?
Dialogue: 0,0:00:26.59,0:00:33.70,Default,,0000,0000,0000,,X is one. Why is 3X is 2?\NWhy is 4 and access three? Why
Dialogue: 0,0:00:33.70,0:00:39.39,Default,,0000,0000,0000,,is 5? Let's see what these\Npoints look like when we put
Dialogue: 0,0:00:39.39,0:00:41.28,Default,,0000,0000,0000,,them on a graph.
Dialogue: 0,0:00:42.23,0:00:44.90,Default,,0000,0000,0000,,The first point, not 2.
Dialogue: 0,0:00:45.68,0:00:46.38,Default,,0000,0000,0000,,Will be here.
Dialogue: 0,0:00:47.69,0:00:50.56,Default,,0000,0000,0000,,An X coordinate of zero\Nand a Y coordinate 2.
Dialogue: 0,0:00:51.89,0:00:56.18,Default,,0000,0000,0000,,The second point 1 three X\Ncoordinate of one Y coordinate
Dialogue: 0,0:00:56.18,0:00:59.47,Default,,0000,0000,0000,,of three. And so on 2 four.
Dialogue: 0,0:01:00.49,0:01:01.72,Default,,0000,0000,0000,,That will be here.
Dialogue: 0,0:01:02.83,0:01:03.87,Default,,0000,0000,0000,,And three 5.
Dialogue: 0,0:01:05.52,0:01:07.28,Default,,0000,0000,0000,,That will be there.
Dialogue: 0,0:01:07.92,0:01:12.65,Default,,0000,0000,0000,,See, we've got four points and\Nvery conveniently we can put a
Dialogue: 0,0:01:12.65,0:01:16.98,Default,,0000,0000,0000,,straight line through them.\NNotice that in every case, the Y
Dialogue: 0,0:01:16.98,0:01:22.89,Default,,0000,0000,0000,,value is always two more than\Nthe X value, so if we add on two
Dialogue: 0,0:01:22.89,0:01:29.20,Default,,0000,0000,0000,,to zero, we get two. If we add\Non 2 to one, we get three, and
Dialogue: 0,0:01:29.20,0:01:35.11,Default,,0000,0000,0000,,so on. the Y value is always the\NX value plus two, so this gives
Dialogue: 0,0:01:35.11,0:01:37.86,Default,,0000,0000,0000,,us the equation of the line the
Dialogue: 0,0:01:37.86,0:01:41.96,Default,,0000,0000,0000,,Y value. Is always the X\Nvalue +2.
Dialogue: 0,0:01:43.25,0:01:47.67,Default,,0000,0000,0000,,Now there are lots and lots of\Nother points on this line, not
Dialogue: 0,0:01:47.67,0:01:51.41,Default,,0000,0000,0000,,just the four that we've\Nplotted, but any point that we
Dialogue: 0,0:01:51.41,0:01:54.81,Default,,0000,0000,0000,,choose on the line will have\Nthis same relationship between
Dialogue: 0,0:01:54.81,0:01:59.57,Default,,0000,0000,0000,,Y&X. the Y value will always be\Nit's X value plus two, so that
Dialogue: 0,0:01:59.57,0:02:03.99,Default,,0000,0000,0000,,is the equation of the line, and\Nvery often we'll label the line
Dialogue: 0,0:02:03.99,0:02:07.05,Default,,0000,0000,0000,,with the equation by writing it\Nalongside like that.
Dialogue: 0,0:02:08.60,0:02:10.92,Default,,0000,0000,0000,,Let's look at some more straight
Dialogue: 0,0:02:10.92,0:02:18.41,Default,,0000,0000,0000,,line graphs. Let's suppose\Nwe start with the
Dialogue: 0,0:02:18.41,0:02:26.31,Default,,0000,0000,0000,,equation Y equals X\Nor drop a table
Dialogue: 0,0:02:26.31,0:02:30.26,Default,,0000,0000,0000,,of values and plot
Dialogue: 0,0:02:30.26,0:02:37.26,Default,,0000,0000,0000,,some points. Again,\Nlet's start with some
Dialogue: 0,0:02:37.26,0:02:43.69,Default,,0000,0000,0000,,X values. Suppose the\NX values run from
Dialogue: 0,0:02:43.69,0:02:46.91,Default,,0000,0000,0000,,012 up to three.
Dialogue: 0,0:02:47.72,0:02:51.82,Default,,0000,0000,0000,,What will the Y value be if the\Nequation is simply Y equals X?
Dialogue: 0,0:02:51.82,0:02:55.63,Default,,0000,0000,0000,,Well, in this case it's a very\Nsimple case. the Y value is
Dialogue: 0,0:02:55.63,0:02:59.44,Default,,0000,0000,0000,,always equal to the X value. So\Nvery simply we can complete the
Dialogue: 0,0:02:59.44,0:03:02.96,Default,,0000,0000,0000,,table. the Y value is always the\Nsame as the X value.
Dialogue: 0,0:03:03.65,0:03:07.47,Default,,0000,0000,0000,,Let's plot these points on the
Dialogue: 0,0:03:07.47,0:03:11.38,Default,,0000,0000,0000,,graph. Access note why is not.
Dialogue: 0,0:03:11.99,0:03:13.33,Default,,0000,0000,0000,,Is the point of the origin.
Dialogue: 0,0:03:14.77,0:03:16.66,Default,,0000,0000,0000,,X is one. Why is one?
Dialogue: 0,0:03:17.24,0:03:18.60,Default,,0000,0000,0000,,Will be here.
Dialogue: 0,0:03:19.14,0:03:21.17,Default,,0000,0000,0000,,And similarly 2233.
Dialogue: 0,0:03:21.68,0:03:24.07,Default,,0000,0000,0000,,Will be there. And there.
Dialogue: 0,0:03:24.93,0:03:27.14,Default,,0000,0000,0000,,So we have a straight line.
Dialogue: 0,0:03:28.32,0:03:30.89,Default,,0000,0000,0000,,Passing through the origin.
Dialogue: 0,0:03:32.40,0:03:35.86,Default,,0000,0000,0000,,Let's ask ourselves a little bit\Nabout the gradient of this line.
Dialogue: 0,0:03:36.48,0:03:40.39,Default,,0000,0000,0000,,Remember to find the gradient of\Nthe line we take, say two points
Dialogue: 0,0:03:40.39,0:03:44.00,Default,,0000,0000,0000,,on it. Let's suppose we take\Nthis point and this point, and
Dialogue: 0,0:03:44.00,0:03:47.92,Default,,0000,0000,0000,,we calculate the change in Y\Ndivided by the change in X. As
Dialogue: 0,0:03:47.92,0:03:50.02,Default,,0000,0000,0000,,we move from one point to the
Dialogue: 0,0:03:50.02,0:03:54.82,Default,,0000,0000,0000,,next. Well, as we move from here\Nto here, why changes from one to
Dialogue: 0,0:03:54.82,0:03:58.41,Default,,0000,0000,0000,,three? So the change in Y is 3 -
Dialogue: 0,0:03:58.41,0:04:03.78,Default,,0000,0000,0000,,1. And the change in X will\Nexchange is from one to three,
Dialogue: 0,0:04:03.78,0:04:07.13,Default,,0000,0000,0000,,so the change in X is also 3 -
Dialogue: 0,0:04:07.13,0:04:12.61,Default,,0000,0000,0000,,1. 3 - 1 is two 3\N- 1 is 2.
Dialogue: 0,0:04:13.26,0:04:15.91,Default,,0000,0000,0000,,So the gradient of this line is
Dialogue: 0,0:04:15.91,0:04:19.33,Default,,0000,0000,0000,,one. Want to write that\Nalongside here? Let's call
Dialogue: 0,0:04:19.33,0:04:22.83,Default,,0000,0000,0000,,it M1. This is the first\Nline of several lines I'm
Dialogue: 0,0:04:22.83,0:04:26.33,Default,,0000,0000,0000,,going to draw. An M1 is\None. The gradient is one.
Dialogue: 0,0:04:27.52,0:04:30.14,Default,,0000,0000,0000,,And also write the equation\Nof the line alongside as
Dialogue: 0,0:04:30.14,0:04:33.02,Default,,0000,0000,0000,,well. So the equation of\Nthis line is why is X?
Dialogue: 0,0:04:35.26,0:04:39.30,Default,,0000,0000,0000,,Let's put another straight line\Non the same graph and this time.
Dialogue: 0,0:04:39.30,0:04:42.34,Default,,0000,0000,0000,,Let's suppose we choose the\Nequation Y equals 2X.
Dialogue: 0,0:04:44.19,0:04:47.13,Default,,0000,0000,0000,,Let's see what the Y\Ncoordinates will be this
Dialogue: 0,0:04:47.13,0:04:50.73,Default,,0000,0000,0000,,time. Well, the Y coordinate\Nis always two times the X
Dialogue: 0,0:04:50.73,0:04:54.00,Default,,0000,0000,0000,,coordinate, so if the X\Ncoordinate is 0, the Y
Dialogue: 0,0:04:54.00,0:04:57.27,Default,,0000,0000,0000,,coordinate will be 2 * 0,\Nwhich is still 0.
Dialogue: 0,0:04:58.78,0:05:01.51,Default,,0000,0000,0000,,When X is one, why will be 2 * 1
Dialogue: 0,0:05:01.51,0:05:06.13,Default,,0000,0000,0000,,which is 2? When X is 2, Y is 2\N* 2, which is 4.
Dialogue: 0,0:05:06.87,0:05:09.19,Default,,0000,0000,0000,,I'm an ex is 3. Why is 2 * 3
Dialogue: 0,0:05:09.19,0:05:14.95,Default,,0000,0000,0000,,which is 6? Let's put these on\Nas well. We've got 00, which is
Dialogue: 0,0:05:14.95,0:05:16.15,Default,,0000,0000,0000,,the origin again.
Dialogue: 0,0:05:17.22,0:05:19.80,Default,,0000,0000,0000,,When X is one way is 2. That's
Dialogue: 0,0:05:19.80,0:05:24.28,Default,,0000,0000,0000,,this point here. When X is 2,\Nwhy is 4?
Dialogue: 0,0:05:25.06,0:05:28.60,Default,,0000,0000,0000,,At this point here, I'm going to\Naccess three wise 6.
Dialogue: 0,0:05:29.14,0:05:34.68,Default,,0000,0000,0000,,She's up there and again we have\Na straight line graph and again.
Dialogue: 0,0:05:35.69,0:05:39.78,Default,,0000,0000,0000,,This line passes through\Nthe origin.
Dialogue: 0,0:05:40.91,0:05:45.03,Default,,0000,0000,0000,,Right, so let's write its\Nequation alongside. Why is 2X?
Dialogue: 0,0:05:45.67,0:05:48.83,Default,,0000,0000,0000,,And let's just think for a\Nminute about the gradient of
Dialogue: 0,0:05:48.83,0:05:51.70,Default,,0000,0000,0000,,this line. Let's take two\Npoints. Let's suppose we take
Dialogue: 0,0:05:51.70,0:05:53.13,Default,,0000,0000,0000,,this point and this point.
Dialogue: 0,0:05:53.91,0:05:55.22,Default,,0000,0000,0000,,The change in Y.
Dialogue: 0,0:05:56.60,0:06:00.11,Default,,0000,0000,0000,,Well, why is changing from two\Nto four? So the changing? Why is
Dialogue: 0,0:06:00.11,0:06:01.73,Default,,0000,0000,0000,,4 takeaway 2 which is 2?
Dialogue: 0,0:06:02.38,0:06:06.52,Default,,0000,0000,0000,,The change in X will exchange is\Nfrom one to two, so the change
Dialogue: 0,0:06:06.52,0:06:10.96,Default,,0000,0000,0000,,in X is just 2 - 1 or one, so\Nthe slope of this line.
Dialogue: 0,0:06:11.52,0:06:13.30,Default,,0000,0000,0000,,Just two.
Dialogue: 0,0:06:14.92,0:06:20.81,Default,,0000,0000,0000,,That's cool that M2 is the slope\Nof the second line, right, M22?
Dialogue: 0,0:06:23.34,0:06:27.55,Default,,0000,0000,0000,,OK, let's do one more. Suppose\Nwe have another equation. And
Dialogue: 0,0:06:27.55,0:06:32.15,Default,,0000,0000,0000,,let's suppose this time the\Nequation is Y equals 3X. So the
Dialogue: 0,0:06:32.15,0:06:37.51,Default,,0000,0000,0000,,Y value is always three times\Nthe X value. We can put these in
Dialogue: 0,0:06:37.51,0:06:41.72,Default,,0000,0000,0000,,straightaway 3 notes and not\N314-3326 and three threes and 9.
Dialogue: 0,0:06:42.28,0:06:44.02,Default,,0000,0000,0000,,And we can plot these on\Nthe same graph.
Dialogue: 0,0:06:45.13,0:06:48.90,Default,,0000,0000,0000,,Again, 00 so the graph is going\Nto pass through the origin.
Dialogue: 0,0:06:50.44,0:06:55.33,Default,,0000,0000,0000,,Two, when X is one, why is\N3 so when X is one? Why is
Dialogue: 0,0:06:55.33,0:06:56.96,Default,,0000,0000,0000,,3 gives me this point?
Dialogue: 0,0:06:58.65,0:07:03.59,Default,,0000,0000,0000,,When X is 2, why is now 6? So\NI've got a point up here and
Dialogue: 0,0:07:03.59,0:07:07.30,Default,,0000,0000,0000,,that's sufficient to to draw in\Nthe straight line and again with
Dialogue: 0,0:07:07.30,0:07:10.70,Default,,0000,0000,0000,,a straight line passing through\Nthe origin is a steeper line
Dialogue: 0,0:07:10.70,0:07:13.42,Default,,0000,0000,0000,,this time. And it's equation is
Dialogue: 0,0:07:13.42,0:07:19.01,Default,,0000,0000,0000,,Y is 3X. So we've got 3 lines\Ndrawn. Now, why is XY is 2, XY
Dialogue: 0,0:07:19.01,0:07:21.09,Default,,0000,0000,0000,,is 3X and all these lines pass
Dialogue: 0,0:07:21.09,0:07:24.98,Default,,0000,0000,0000,,through the origin? Let's just\Nget the gradient of this line or
Dialogue: 0,0:07:24.98,0:07:28.02,Default,,0000,0000,0000,,the gradient of this line.\NAgain. Let's take two points on
Dialogue: 0,0:07:28.02,0:07:32.40,Default,,0000,0000,0000,,it. The change in Y going from\Nthis point to this point. Well,
Dialogue: 0,0:07:32.40,0:07:36.62,Default,,0000,0000,0000,,why is changing from 3 up to\Nsix? So the change in Y is 6?
Dialogue: 0,0:07:36.62,0:07:37.74,Default,,0000,0000,0000,,Subtract 3 or three.
Dialogue: 0,0:07:38.45,0:07:42.74,Default,,0000,0000,0000,,And the change in XLX is\Nchanging from one to two.
Dialogue: 0,0:07:43.25,0:07:47.82,Default,,0000,0000,0000,,So the change in access 2 - 1,\Nwhich is just one.
Dialogue: 0,0:07:47.83,0:07:52.53,Default,,0000,0000,0000,,So the gradient this time is 3.\NLet's label that and three.
Dialogue: 0,0:07:53.34,0:07:56.63,Default,,0000,0000,0000,,Now this is no coincidence.\NYou'll notice that in every
Dialogue: 0,0:07:56.63,0:08:01.24,Default,,0000,0000,0000,,case, the gradient in this case\N3 is the same as the number that
Dialogue: 0,0:08:01.24,0:08:05.18,Default,,0000,0000,0000,,is multiplying the X in the\Nequation. Same is true here. The
Dialogue: 0,0:08:05.18,0:08:09.13,Default,,0000,0000,0000,,gradient is 2, which is the\Nnumber. Multiplying the X in the
Dialogue: 0,0:08:09.13,0:08:13.08,Default,,0000,0000,0000,,equation. And again here. Why is\NX the number? Multiplying X is
Dialogue: 0,0:08:13.08,0:08:15.05,Default,,0000,0000,0000,,one and the gradient is one.
Dialogue: 0,0:08:15.77,0:08:20.44,Default,,0000,0000,0000,,Now we deduce from this a\Ngeneral result that whenever we
Dialogue: 0,0:08:20.44,0:08:25.54,Default,,0000,0000,0000,,have an equation of the form Y\Nequals MX. What this represents
Dialogue: 0,0:08:25.54,0:08:27.24,Default,,0000,0000,0000,,is a straight line.
Dialogue: 0,0:08:28.32,0:08:34.11,Default,,0000,0000,0000,,It's a line which is passing\Nthrough the origin.
Dialogue: 0,0:08:34.11,0:08:40.92,Default,,0000,0000,0000,,And it's gradient is\NM. The number multiplying
Dialogue: 0,0:08:40.92,0:08:44.32,Default,,0000,0000,0000,,the X is the
Dialogue: 0,0:08:44.32,0:08:47.99,Default,,0000,0000,0000,,gradient. That's a very\Nimportant result, it's well
Dialogue: 0,0:08:47.99,0:08:52.32,Default,,0000,0000,0000,,worth remembering that whenever\Nyou see why is a constant M
Dialogue: 0,0:08:52.32,0:08:57.05,Default,,0000,0000,0000,,Times X will be a straight line\Nwill be passing through the
Dialogue: 0,0:08:57.05,0:09:01.38,Default,,0000,0000,0000,,origin and the gradient will be\Nthe number that's multiplying X.
Dialogue: 0,0:09:02.59,0:09:09.23,Default,,0000,0000,0000,,Let's have a\Nlook at some
Dialogue: 0,0:09:09.23,0:09:12.55,Default,,0000,0000,0000,,other equations of
Dialogue: 0,0:09:12.55,0:09:19.79,Default,,0000,0000,0000,,straight lines. Let's\Nhave a look at Y equals 2X
Dialogue: 0,0:09:19.79,0:09:24.43,Default,,0000,0000,0000,,plus one. Very similar to the\None we had before, but now I've
Dialogue: 0,0:09:24.43,0:09:27.97,Default,,0000,0000,0000,,added on a number at the end\Nhere. Let's choose some X and
Dialogue: 0,0:09:27.97,0:09:35.32,Default,,0000,0000,0000,,some Y values. When\NX is
Dialogue: 0,0:09:35.32,0:09:41.91,Default,,0000,0000,0000,,0. Why will be\N2 * 0 which is 0 plus one? So
Dialogue: 0,0:09:41.91,0:09:43.82,Default,,0000,0000,0000,,when X is 0 while B1.
Dialogue: 0,0:09:44.67,0:09:49.25,Default,,0000,0000,0000,,When X is one 2, one or two plus\None gives you 3.
Dialogue: 0,0:09:50.25,0:09:54.76,Default,,0000,0000,0000,,And when X is 222 to four and\None is 5, so with those three
Dialogue: 0,0:09:54.76,0:09:58.68,Default,,0000,0000,0000,,points we can plot a graph when\NX is not. Why is one?
Dialogue: 0,0:09:59.32,0:10:00.30,Default,,0000,0000,0000,,It's there.
Dialogue: 0,0:10:01.36,0:10:05.00,Default,,0000,0000,0000,,When X is one, why is 3? So we\Ncome up to here.
Dialogue: 0,0:10:05.76,0:10:09.36,Default,,0000,0000,0000,,I'm in access 2. Why is 5 which\Ntakes us up to there?
Dialogue: 0,0:10:10.02,0:10:15.54,Default,,0000,0000,0000,,And there's my straight line\Ngraph through those points.
Dialogue: 0,0:10:17.14,0:10:22.00,Default,,0000,0000,0000,,Not label it. Y equals 2X\Nplus one.
Dialogue: 0,0:10:24.22,0:10:29.37,Default,,0000,0000,0000,,Let's have another one. Suppose\Nwe have Y equals 2 X +4.
Dialogue: 0,0:10:30.69,0:10:32.04,Default,,0000,0000,0000,,Let's see what happens this
Dialogue: 0,0:10:32.04,0:10:39.78,Default,,0000,0000,0000,,time. Let's suppose we\Nstart with a negative
Dialogue: 0,0:10:39.78,0:10:41.73,Default,,0000,0000,0000,,X value.
Dialogue: 0,0:10:43.03,0:10:46.88,Default,,0000,0000,0000,,Access minus one. What will the\NY value be? Effects is minus,
Dialogue: 0,0:10:46.88,0:10:49.13,Default,,0000,0000,0000,,one will get two times minus one
Dialogue: 0,0:10:49.13,0:10:53.41,Default,,0000,0000,0000,,is minus 2. And 4 - 2\Nis 2.
Dialogue: 0,0:10:56.04,0:11:01.74,Default,,0000,0000,0000,,Let's choose X to be 0 when X is\Nzero, will get 2 zeros as O plus
Dialogue: 0,0:11:01.74,0:11:06.76,Default,,0000,0000,0000,,454. So ex is one we get 2 ones\Nor 2 + 4 is 6.
Dialogue: 0,0:11:07.42,0:11:08.56,Default,,0000,0000,0000,,Let's put those points.
Dialogue: 0,0:11:09.19,0:11:11.40,Default,,0000,0000,0000,,So if X is minus one, why is 2?
Dialogue: 0,0:11:13.87,0:11:15.90,Default,,0000,0000,0000,,If X is zero, why is 4?
Dialogue: 0,0:11:18.42,0:11:22.36,Default,,0000,0000,0000,,And effects is one. Why is 6,\Nwhich is a point of the.
Dialogue: 0,0:11:23.23,0:11:28.11,Default,,0000,0000,0000,,That's the line Y equals 2X\Nplus four, and you'll notice
Dialogue: 0,0:11:28.11,0:11:33.44,Default,,0000,0000,0000,,from looking at it that the\Ntwo lines that we have now
Dialogue: 0,0:11:33.44,0:11:37.44,Default,,0000,0000,0000,,drawn a parallel, and that's\Nprecisely because they've got
Dialogue: 0,0:11:37.44,0:11:40.55,Default,,0000,0000,0000,,the same gradient. The number\Nmultiplying X.
Dialogue: 0,0:11:42.31,0:11:46.70,Default,,0000,0000,0000,,Let's look at one more. Why is\N2X minus one?
Dialogue: 0,0:11:47.25,0:11:49.17,Default,,0000,0000,0000,,Again, let's have some X values.
Dialogue: 0,0:11:49.72,0:11:53.16,Default,,0000,0000,0000,,And some Y values supposing X
Dialogue: 0,0:11:53.16,0:11:57.99,Default,,0000,0000,0000,,is 0. Well, if X is zero and why\Nis 2X minus one?
Dialogue: 0,0:11:58.51,0:12:02.65,Default,,0000,0000,0000,,The Y value will be 2 notes and\Nnot subtract. 1 is minus one.
Dialogue: 0,0:12:03.86,0:12:09.38,Default,,0000,0000,0000,,If X is, one will get 2 ones, or\Ntwo. Subtract 1 is plus one.
Dialogue: 0,0:12:10.49,0:12:11.85,Default,,0000,0000,0000,,And effects is 2.
Dialogue: 0,0:12:12.35,0:12:16.66,Default,,0000,0000,0000,,Two tubes of 4 - 1 is 3. Again,\Nwe've got three points. That's
Dialogue: 0,0:12:16.66,0:12:20.36,Default,,0000,0000,0000,,plenty points to put on the\Ngraph. Effects is not. Why is
Dialogue: 0,0:12:20.36,0:12:23.98,Default,,0000,0000,0000,,minus one? Thanks is not\Nwise minus one gives me a
Dialogue: 0,0:12:23.98,0:12:24.78,Default,,0000,0000,0000,,point down here.
Dialogue: 0,0:12:27.04,0:12:28.10,Default,,0000,0000,0000,,If X is one.
Dialogue: 0,0:12:28.69,0:12:30.93,Default,,0000,0000,0000,,Why is one gives\Nme a point here?
Dialogue: 0,0:12:32.34,0:12:35.72,Default,,0000,0000,0000,,And if X is 2 wise, three gives\Nme that point there.
Dialogue: 0,0:12:39.93,0:12:46.24,Default,,0000,0000,0000,,And there's the straight line Y\Nequals 2X minus one, and again
Dialogue: 0,0:12:46.24,0:12:47.82,Default,,0000,0000,0000,,this third line.
Dialogue: 0,0:12:48.54,0:12:51.97,Default,,0000,0000,0000,,Is parallel to the previous two\Nlines and it's parallel because
Dialogue: 0,0:12:51.97,0:12:55.40,Default,,0000,0000,0000,,it's got the same gradient and\Nit's got the same gradient
Dialogue: 0,0:12:55.40,0:12:59.15,Default,,0000,0000,0000,,because in every case we've got\N2X the number. Multiplying X is
Dialogue: 0,0:12:59.15,0:13:03.46,Default,,0000,0000,0000,,the same. So what's different\Nabout the lines? Well, what is
Dialogue: 0,0:13:03.46,0:13:05.01,Default,,0000,0000,0000,,different is that they're all
Dialogue: 0,0:13:05.01,0:13:08.32,Default,,0000,0000,0000,,cutting. The Y axis\Nat a different point.
Dialogue: 0,0:13:09.54,0:13:13.78,Default,,0000,0000,0000,,This line is cutting the Y axis\Nat the point where. Why is 4?
Dialogue: 0,0:13:14.94,0:13:17.29,Default,,0000,0000,0000,,Note that the number 4\Nappears in the equation.
Dialogue: 0,0:13:18.93,0:13:23.96,Default,,0000,0000,0000,,This line. Cuts the Y axis when\NY is one, and again one appears
Dialogue: 0,0:13:23.96,0:13:29.04,Default,,0000,0000,0000,,in the equation. And again, this\Nline cuts the Y axis at minus
Dialogue: 0,0:13:29.04,0:13:33.29,Default,,0000,0000,0000,,one and minus one appears in the\Nequation, and this gives us a
Dialogue: 0,0:13:33.29,0:13:37.54,Default,,0000,0000,0000,,general rule. If we have an\Nequation of the form Y equals MX
Dialogue: 0,0:13:37.54,0:13:42.45,Default,,0000,0000,0000,,plus C, the number that is on\Nits own at the end. Here the C
Dialogue: 0,0:13:42.45,0:13:46.70,Default,,0000,0000,0000,,which was the four or the one or\Nthe minus one, tells us
Dialogue: 0,0:13:46.70,0:13:48.66,Default,,0000,0000,0000,,whereabouts on the Y axis that
Dialogue: 0,0:13:48.66,0:13:51.75,Default,,0000,0000,0000,,the graph cuts. And we call this
Dialogue: 0,0:13:51.75,0:13:57.52,Default,,0000,0000,0000,,value. Either the for their or\Nthe one there, or the minus one
Dialogue: 0,0:13:57.52,0:14:02.59,Default,,0000,0000,0000,,there. We call that the vertical\Nintercept so the value of C is
Dialogue: 0,0:14:02.59,0:14:03.76,Default,,0000,0000,0000,,the vertical intercept.
Dialogue: 0,0:14:03.84,0:14:09.98,Default,,0000,0000,0000,,So now whenever you see\Nan equation of the form
Dialogue: 0,0:14:09.98,0:14:15.51,Default,,0000,0000,0000,,Y equals a number times\NX plus another number.
Dialogue: 0,0:14:15.51,0:14:19.19,Default,,0000,0000,0000,,So why equals MX plus C?
Dialogue: 0,0:14:20.33,0:14:22.20,Default,,0000,0000,0000,,That represents a straight line
Dialogue: 0,0:14:22.20,0:14:24.90,Default,,0000,0000,0000,,graph. Where M is the gradient
Dialogue: 0,0:14:24.90,0:14:29.19,Default,,0000,0000,0000,,of the line. And sees the\Nvertical intercept, which is the
Dialogue: 0,0:14:29.19,0:14:31.95,Default,,0000,0000,0000,,place where the graph crosses\Nthe vertical axis.
Dialogue: 0,0:14:33.48,0:14:39.90,Default,,0000,0000,0000,,Now sometimes when we get the\Nequation of a straight line, it
Dialogue: 0,0:14:39.90,0:14:46.32,Default,,0000,0000,0000,,doesn't always appear in the\Nform Y equals MX plus C. Let
Dialogue: 0,0:14:46.32,0:14:48.100,Default,,0000,0000,0000,,me give you an example.
Dialogue: 0,0:14:49.00,0:14:53.90,Default,,0000,0000,0000,,Let's consider this equation 3.\NY minus two X equals 6. Now at
Dialogue: 0,0:14:53.90,0:14:58.80,Default,,0000,0000,0000,,first sight that doesn't look as\Nthough it's in the form Y equals
Dialogue: 0,0:14:58.80,0:15:02.57,Default,,0000,0000,0000,,MX plus C which is our\Nrecognisable form of the
Dialogue: 0,0:15:02.57,0:15:08.23,Default,,0000,0000,0000,,equation of a straight line. But\Nwhat we can do is we can do some
Dialogue: 0,0:15:08.23,0:15:12.75,Default,,0000,0000,0000,,algebraic manipulation on this\Nto try to write it in this form
Dialogue: 0,0:15:12.75,0:15:18.03,Default,,0000,0000,0000,,and one of the advantages of\Ndoing that is that if we can get
Dialogue: 0,0:15:18.03,0:15:19.54,Default,,0000,0000,0000,,it into this form.
Dialogue: 0,0:15:19.54,0:15:23.18,Default,,0000,0000,0000,,We can read off what the\Ngradients and the vertical
Dialogue: 0,0:15:23.18,0:15:25.36,Default,,0000,0000,0000,,intercept are, so let's work on
Dialogue: 0,0:15:25.36,0:15:30.41,Default,,0000,0000,0000,,this. I'll start by adding 2X\Nto both sides.
Dialogue: 0,0:15:30.41,0:15:35.37,Default,,0000,0000,0000,,To remove this minus 2X from\Nhere. So if we add 2X to both
Dialogue: 0,0:15:35.37,0:15:37.84,Default,,0000,0000,0000,,sides will get 2X plus six on
Dialogue: 0,0:15:37.84,0:15:43.42,Default,,0000,0000,0000,,the right. And now if I divide\Nboth sides by three, I'll get Y
Dialogue: 0,0:15:43.42,0:15:45.92,Default,,0000,0000,0000,,on its own, which is what I'm
Dialogue: 0,0:15:45.92,0:15:51.69,Default,,0000,0000,0000,,looking for. Dividing 2X by\Nthree gives me why is 2/3 of
Dialogue: 0,0:15:51.69,0:15:57.23,Default,,0000,0000,0000,,X? And if I divide 6 by\Nthree, I'll get 2.
Dialogue: 0,0:15:57.24,0:16:02.38,Default,,0000,0000,0000,,Now this is a much more familiar\Nform. This is of the form Y
Dialogue: 0,0:16:02.38,0:16:07.52,Default,,0000,0000,0000,,equals MX plus. See where we can\Nread off the gradient M is 2/3
Dialogue: 0,0:16:07.52,0:16:09.35,Default,,0000,0000,0000,,and the vertical intercept see
Dialogue: 0,0:16:09.35,0:16:13.64,Default,,0000,0000,0000,,is 2. So be aware that sometimes\Nan equation that you see might
Dialogue: 0,0:16:13.64,0:16:16.34,Default,,0000,0000,0000,,not at first sight look as\Nthough it's a straight line
Dialogue: 0,0:16:16.34,0:16:19.77,Default,,0000,0000,0000,,equation, but by doing some work\Non it you can get it into a
Dialogue: 0,0:16:19.77,0:16:25.60,Default,,0000,0000,0000,,recognizable form. About\Nanother
Dialogue: 0,0:16:25.60,0:16:31.02,Default,,0000,0000,0000,,example. Suppose we're given\Nsome information about a
Dialogue: 0,0:16:31.02,0:16:36.69,Default,,0000,0000,0000,,straight line graph, and we want\Nto try and find out what the
Dialogue: 0,0:16:36.69,0:16:41.05,Default,,0000,0000,0000,,equation is. So, for example,\Nsuppose we're told that a
Dialogue: 0,0:16:41.05,0:16:45.85,Default,,0000,0000,0000,,straight line has gradient, a\Nfifth and were told also that
Dialogue: 0,0:16:45.85,0:16:48.03,Default,,0000,0000,0000,,it's vertical intercept. See is
Dialogue: 0,0:16:48.03,0:16:51.35,Default,,0000,0000,0000,,one. Let's see if we can\Nwrite down the equation.
Dialogue: 0,0:16:52.50,0:16:56.21,Default,,0000,0000,0000,,Well, we know that a\Nstraight line has equation
Dialogue: 0,0:16:56.21,0:16:58.27,Default,,0000,0000,0000,,Y equals MX plus C.
Dialogue: 0,0:16:59.62,0:17:05.22,Default,,0000,0000,0000,,So we can substitute are known\Nvalues in M is going to be 1/5.
Dialogue: 0,0:17:05.22,0:17:11.22,Default,,0000,0000,0000,,See is going to be one. So our\Nequation is Y equals 1/5 of X
Dialogue: 0,0:17:11.22,0:17:14.02,Default,,0000,0000,0000,,plus one Y equals MX plus C.
Dialogue: 0,0:17:14.77,0:17:18.72,Default,,0000,0000,0000,,Now we might not always choose\Nto leave it in that form, so let
Dialogue: 0,0:17:18.72,0:17:22.10,Default,,0000,0000,0000,,me just show you how else we\Nmight write it. There's a
Dialogue: 0,0:17:22.10,0:17:25.20,Default,,0000,0000,0000,,fraction here of the 5th, and if\Nwe multiply everything through
Dialogue: 0,0:17:25.20,0:17:28.31,Default,,0000,0000,0000,,by 5, we can remove this\Nfraction. So let's multiply both
Dialogue: 0,0:17:28.31,0:17:31.97,Default,,0000,0000,0000,,sides by 5, will get 5 Y the\Nfiles or cancel. When we
Dialogue: 0,0:17:31.97,0:17:33.95,Default,,0000,0000,0000,,multiply by 5 here just to leave
Dialogue: 0,0:17:33.95,0:17:36.81,Default,,0000,0000,0000,,X. And five ones of five.
Dialogue: 0,0:17:37.62,0:17:42.69,Default,,0000,0000,0000,,So this form is equivalent to\Nthis form, but just written in a
Dialogue: 0,0:17:42.69,0:17:46.62,Default,,0000,0000,0000,,different way. We could\Nrearrange it again just by
Dialogue: 0,0:17:46.62,0:17:50.52,Default,,0000,0000,0000,,bringing everything over to the\Nleft hand side, so we might
Dialogue: 0,0:17:50.52,0:17:55.84,Default,,0000,0000,0000,,write 5 Y minus X. Minus five is\Nnot, so that is another form of
Dialogue: 0,0:17:55.84,0:17:59.75,Default,,0000,0000,0000,,the same equation and we'll see\Nsome equations written in this
Dialogue: 0,0:17:59.75,0:18:01.17,Default,,0000,0000,0000,,form which later on.
Dialogue: 0,0:18:02.41,0:18:09.31,Default,,0000,0000,0000,,OK, let's have\Na look at
Dialogue: 0,0:18:09.31,0:18:14.35,Default,,0000,0000,0000,,another example. Suppose now\Nwe're interested in trying to
Dialogue: 0,0:18:14.35,0:18:17.82,Default,,0000,0000,0000,,find the equation of a line\Nwhich has a gradient of 1/3.
Dialogue: 0,0:18:18.96,0:18:22.47,Default,,0000,0000,0000,,And this time, instead of being\Ngiven the vertical intercept,
Dialogue: 0,0:18:22.47,0:18:25.98,Default,,0000,0000,0000,,we're going to be given some\Ninformation about a point
Dialogue: 0,0:18:25.98,0:18:30.19,Default,,0000,0000,0000,,through which the line passes.\NSo suppose that we know that the
Dialogue: 0,0:18:30.19,0:18:33.00,Default,,0000,0000,0000,,line passes through the points\Nwith coordinates 12.
Dialogue: 0,0:18:33.80,0:18:36.37,Default,,0000,0000,0000,,Let's see if we can figure\Nout what the equation of
Dialogue: 0,0:18:36.37,0:18:37.08,Default,,0000,0000,0000,,the line is.
Dialogue: 0,0:18:39.09,0:18:44.51,Default,,0000,0000,0000,,Start with our general form Y\Nequals MX plus C and we put in
Dialogue: 0,0:18:44.51,0:18:46.44,Default,,0000,0000,0000,,the information that we already
Dialogue: 0,0:18:46.44,0:18:52.57,Default,,0000,0000,0000,,know. We know that the gradient\NM is 1/3, so we can put that in
Dialogue: 0,0:18:52.57,0:18:53.66,Default,,0000,0000,0000,,here straight away.
Dialogue: 0,0:18:53.68,0:18:57.31,Default,,0000,0000,0000,,We don't know the vertical\Nintercept. We're going to have
Dialogue: 0,0:18:57.31,0:19:00.21,Default,,0000,0000,0000,,to do a bit of work to find
Dialogue: 0,0:19:00.21,0:19:05.26,Default,,0000,0000,0000,,that. But what we do know is\Nthat the line passes through
Dialogue: 0,0:19:05.26,0:19:10.48,Default,,0000,0000,0000,,this point. What that means is\Nthat when X is one, why has the
Dialogue: 0,0:19:10.48,0:19:14.59,Default,,0000,0000,0000,,value 2? And we can use that\Ninformation in this equation.
Dialogue: 0,0:19:15.38,0:19:17.38,Default,,0000,0000,0000,,So we're going to put, why is
Dialogue: 0,0:19:17.38,0:19:23.62,Default,,0000,0000,0000,,2IN. X is one in home, 3\Nthird times, one is just a
Dialogue: 0,0:19:23.62,0:19:28.88,Default,,0000,0000,0000,,third. Let's see from this we\Ncan workout what Sears.
Dialogue: 0,0:19:29.44,0:19:33.97,Default,,0000,0000,0000,,So two is the same as 6 thirds\Nand if we take a third off, both
Dialogue: 0,0:19:33.97,0:19:38.21,Default,,0000,0000,0000,,sides will have 5 thirds is see\Nso you can see we can use the
Dialogue: 0,0:19:38.21,0:19:39.91,Default,,0000,0000,0000,,information about a point on the
Dialogue: 0,0:19:39.91,0:19:44.45,Default,,0000,0000,0000,,line. To find the vertical\Nintercept, see so. Now we know
Dialogue: 0,0:19:44.45,0:19:48.18,Default,,0000,0000,0000,,everything about this line. We\Nknow it's vertical intercept and
Dialogue: 0,0:19:48.18,0:19:53.03,Default,,0000,0000,0000,,we know its gradient. So the\Nequation of the line is why is
Dialogue: 0,0:19:53.03,0:19:54.52,Default,,0000,0000,0000,,1/3 X +5 thirds?
Dialogue: 0,0:19:54.53,0:20:00.90,Default,,0000,0000,0000,,I want to do that again, but I\Nwant to do it for more general
Dialogue: 0,0:20:00.90,0:20:05.16,Default,,0000,0000,0000,,case where we haven't got\Nspecific values for the gradient
Dialogue: 0,0:20:05.16,0:20:09.83,Default,,0000,0000,0000,,and we haven't got specific\Nvalues for the point. So this
Dialogue: 0,0:20:09.83,0:20:14.50,Default,,0000,0000,0000,,time, let's suppose we've got a\Nstraight line. This gradient is
Dialogue: 0,0:20:14.50,0:20:20.16,Default,,0000,0000,0000,,M. But it passes through a\Npoint with arbitrary coordinates
Dialogue: 0,0:20:20.16,0:20:21.59,Default,,0000,0000,0000,,X one. I want.
Dialogue: 0,0:20:22.25,0:20:24.90,Default,,0000,0000,0000,,Let's see if we can\Nfind a formula for the
Dialogue: 0,0:20:24.90,0:20:25.96,Default,,0000,0000,0000,,equation of the line.
Dialogue: 0,0:20:27.62,0:20:31.68,Default,,0000,0000,0000,,Always go back to what we know.\NWe know already that any
Dialogue: 0,0:20:31.68,0:20:35.06,Default,,0000,0000,0000,,straight line has this equation.\NY is MX plus C.
Dialogue: 0,0:20:35.07,0:20:40.23,Default,,0000,0000,0000,,What do we know what we're told\Nthe gradient is M so that we can
Dialogue: 0,0:20:40.23,0:20:43.43,Default,,0000,0000,0000,,leave alone. But we don't know\Nthe vertical intercept. See
Dialogue: 0,0:20:43.43,0:20:45.04,Default,,0000,0000,0000,,let's see if we can find it.
Dialogue: 0,0:20:46.24,0:20:49.70,Default,,0000,0000,0000,,Use what we do now. We do know\Nthat the line passes through
Dialogue: 0,0:20:49.70,0:20:54.66,Default,,0000,0000,0000,,this point. So that we know that\Nwhen X has the value X one.
Dialogue: 0,0:20:55.53,0:20:57.26,Default,,0000,0000,0000,,Why has the value? Why one?
Dialogue: 0,0:20:57.88,0:20:59.32,Default,,0000,0000,0000,,So I'm going to put those values
Dialogue: 0,0:20:59.32,0:21:06.44,Default,,0000,0000,0000,,in here. So why has the value?\NWhy one when X has the value
Dialogue: 0,0:21:06.44,0:21:13.12,Default,,0000,0000,0000,,X one? See, now we can rearrange\Nthis to find C. So take the MX
Dialogue: 0,0:21:13.12,0:21:14.86,Default,,0000,0000,0000,,one off both sides.
Dialogue: 0,0:21:15.74,0:21:20.44,Default,,0000,0000,0000,,That will give me the value for\NC and this value for see that
Dialogue: 0,0:21:20.44,0:21:24.48,Default,,0000,0000,0000,,we have found, which you\Nrealize now is made up of. Only
Dialogue: 0,0:21:24.48,0:21:29.85,Default,,0000,0000,0000,,the things we knew. We knew the\NM we knew the X one and Y one.
Dialogue: 0,0:21:29.85,0:21:34.56,Default,,0000,0000,0000,,So in fact we know this value.\NNow we put this value back into
Dialogue: 0,0:21:34.56,0:21:38.92,Default,,0000,0000,0000,,the general equation so will\Nhave Y equals MX plus C and see
Dialogue: 0,0:21:38.92,0:21:40.27,Default,,0000,0000,0000,,now is all this.
Dialogue: 0,0:21:41.61,0:21:45.79,Default,,0000,0000,0000,,And that is the equation of a\Nline with gradient M passing
Dialogue: 0,0:21:45.79,0:21:50.31,Default,,0000,0000,0000,,through X one Y one. We don't\Nnormally leave it in this form.
Dialogue: 0,0:21:50.31,0:21:53.79,Default,,0000,0000,0000,,We write it in a slightly\Ndifferent way. It's usually
Dialogue: 0,0:21:53.79,0:21:57.97,Default,,0000,0000,0000,,written like this. We subtract\Nwhy one off both sides to give
Dialogue: 0,0:21:57.97,0:22:00.75,Default,,0000,0000,0000,,us Y, minus Y one, and that will
Dialogue: 0,0:22:00.75,0:22:06.63,Default,,0000,0000,0000,,disappear. And we factorize the\NMX and the NX one by taking out
Dialogue: 0,0:22:06.63,0:22:12.39,Default,,0000,0000,0000,,the common factor of M will be\Nleft with X and minus X one.
Dialogue: 0,0:22:13.29,0:22:18.36,Default,,0000,0000,0000,,And that is an important result,\Nbecause this formula gives us
Dialogue: 0,0:22:18.36,0:22:20.67,Default,,0000,0000,0000,,the equation of a line.
Dialogue: 0,0:22:21.49,0:22:25.27,Default,,0000,0000,0000,,With gradient M and which passes\Nthrough a point where the X
Dialogue: 0,0:22:25.27,0:22:28.74,Default,,0000,0000,0000,,coordinate is X one and the Y\Ncoordinate is why one?
Dialogue: 0,0:22:29.33,0:22:36.48,Default,,0000,0000,0000,,Let's look at\Na specific example.
Dialogue: 0,0:22:36.65,0:22:41.61,Default,,0000,0000,0000,,Suppose we're interested in a\Nstraight line where the gradient
Dialogue: 0,0:22:41.61,0:22:46.57,Default,,0000,0000,0000,,is minus 2, and it passes\Nthrough the point with
Dialogue: 0,0:22:46.57,0:22:48.55,Default,,0000,0000,0000,,coordinates minus 3 two.
Dialogue: 0,0:22:50.42,0:22:55.56,Default,,0000,0000,0000,,We know the general form of a\Nstraight line, it's why minus
Dialogue: 0,0:22:55.56,0:23:00.69,Default,,0000,0000,0000,,why one is MX minus X one?\NThat's our general results and
Dialogue: 0,0:23:00.69,0:23:05.83,Default,,0000,0000,0000,,all we need to do is put this\Ninformation into this formula.
Dialogue: 0,0:23:06.69,0:23:11.08,Default,,0000,0000,0000,,Why one is the Y coordinate\Ncoordinate of the known point
Dialogue: 0,0:23:11.08,0:23:12.28,Default,,0000,0000,0000,,which is 2?
Dialogue: 0,0:23:12.45,0:23:14.44,Default,,0000,0000,0000,,M is the gradient which is minus
Dialogue: 0,0:23:14.44,0:23:20.00,Default,,0000,0000,0000,,2. X minus X one is the X\Ncoordinates of the known point,
Dialogue: 0,0:23:20.00,0:23:21.47,Default,,0000,0000,0000,,which is minus 3.
Dialogue: 0,0:23:22.02,0:23:27.77,Default,,0000,0000,0000,,So tidying this up, we've got Y\Nminus two is going to be minus
Dialogue: 0,0:23:27.77,0:23:33.12,Default,,0000,0000,0000,,2X plus three, and if we remove\Nthe brackets, Y minus two is
Dialogue: 0,0:23:33.12,0:23:35.17,Default,,0000,0000,0000,,minus 2 X minus 6.
Dialogue: 0,0:23:36.38,0:23:40.79,Default,,0000,0000,0000,,And finally, if we add two to\Nboth sides, we shall get why is
Dialogue: 0,0:23:40.79,0:23:44.57,Default,,0000,0000,0000,,minus 2 X minus four, and that's\Nthe equation of the straight
Dialogue: 0,0:23:44.57,0:23:47.72,Default,,0000,0000,0000,,line with gradient minus 2\Npassing through this point. And
Dialogue: 0,0:23:47.72,0:23:51.82,Default,,0000,0000,0000,,there's always a check you can\Napply because we can look at the
Dialogue: 0,0:23:51.82,0:23:55.60,Default,,0000,0000,0000,,final equation we've got and we\Ncan observe from here that the
Dialogue: 0,0:23:55.60,0:23:57.17,Default,,0000,0000,0000,,gradient is indeed minus 2.
Dialogue: 0,0:23:57.80,0:24:01.71,Default,,0000,0000,0000,,And we can also pop in an X\Nvalue of minus three into here.
Dialogue: 0,0:24:02.42,0:24:06.93,Default,,0000,0000,0000,,Minus two times minus three is\Nplus six and six takeaway four
Dialogue: 0,0:24:06.93,0:24:10.69,Default,,0000,0000,0000,,is 2 and that's the\Ncorresponding why value? So this
Dialogue: 0,0:24:10.69,0:24:12.95,Default,,0000,0000,0000,,built in checks that you can
Dialogue: 0,0:24:12.95,0:24:15.89,Default,,0000,0000,0000,,apply. Let's have a look at\Nanother slightly different
Dialogue: 0,0:24:15.89,0:24:19.71,Default,,0000,0000,0000,,example, and in this example\NI'm not going to give you the
Dialogue: 0,0:24:19.71,0:24:22.57,Default,,0000,0000,0000,,gradient of the line.\NInstead, we're going to have
Dialogue: 0,0:24:22.57,0:24:26.07,Default,,0000,0000,0000,,two points on the line. So\Nlet's suppose are two points
Dialogue: 0,0:24:26.07,0:24:30.20,Default,,0000,0000,0000,,are minus 1, two and two 4,\Nso we don't know the gradient
Dialogue: 0,0:24:30.20,0:24:33.07,Default,,0000,0000,0000,,and we don't know the\Nvertical intercept. We just
Dialogue: 0,0:24:33.07,0:24:36.88,Default,,0000,0000,0000,,know two points on the line,\Nand we've got to try to
Dialogue: 0,0:24:36.88,0:24:39.43,Default,,0000,0000,0000,,determine what the equation\Nof the line is.
Dialogue: 0,0:24:40.66,0:24:42.45,Default,,0000,0000,0000,,Now let's see how we can do
Dialogue: 0,0:24:42.45,0:24:47.20,Default,,0000,0000,0000,,this. One thing we can do is we\Ncan calculate the gradient of
Dialogue: 0,0:24:47.20,0:24:50.69,Default,,0000,0000,0000,,the line because we know how to\Ncalculate the gradient of the
Dialogue: 0,0:24:50.69,0:24:51.86,Default,,0000,0000,0000,,line joining two points.
Dialogue: 0,0:24:52.38,0:24:57.08,Default,,0000,0000,0000,,So let's do that. First of all,\Nthe gradient of the line will be
Dialogue: 0,0:24:57.08,0:25:00.78,Default,,0000,0000,0000,,the difference in the Y\Ncoordinates, which is 4 - 2.
Dialogue: 0,0:25:01.20,0:25:06.86,Default,,0000,0000,0000,,Divided by the difference in the\NX coordinates, which is 2 minus
Dialogue: 0,0:25:06.86,0:25:13.71,Default,,0000,0000,0000,,minus one. 4 - 2 is 2 and 2\Nminus minus one is 2 + 1 which
Dialogue: 0,0:25:13.71,0:25:17.59,Default,,0000,0000,0000,,is 3. So the gradient of this\Nline is 2/3.
Dialogue: 0,0:25:18.66,0:25:23.05,Default,,0000,0000,0000,,Now we know the gradients and we\Nknow at least one point through
Dialogue: 0,0:25:23.05,0:25:27.45,Default,,0000,0000,0000,,which the line passes. Because\Nwe know two. So we can use the
Dialogue: 0,0:25:27.45,0:25:32.18,Default,,0000,0000,0000,,previous formula Y minus Y one\Nis MX minus X one. So that's the
Dialogue: 0,0:25:32.18,0:25:33.19,Default,,0000,0000,0000,,formula we use.
Dialogue: 0,0:25:33.93,0:25:37.72,Default,,0000,0000,0000,,Why minus why? One doesn't\Nmatter which of the two points
Dialogue: 0,0:25:37.72,0:25:39.80,Default,,0000,0000,0000,,we take? Let's take the .24.
Dialogue: 0,0:25:40.78,0:25:44.35,Default,,0000,0000,0000,,So the Y coordinate is 4.
Dialogue: 0,0:25:44.45,0:25:47.65,Default,,0000,0000,0000,,M we found is 2/3.
Dialogue: 0,0:25:47.65,0:25:52.29,Default,,0000,0000,0000,,X minus the X coordinate, which\Nis 2. So that's our equation of
Dialogue: 0,0:25:52.29,0:25:57.65,Default,,0000,0000,0000,,the line and if we wanted to do\Nwe can tidy this up a little
Dialogue: 0,0:25:57.65,0:26:03.36,Default,,0000,0000,0000,,bit. Y minus four is 2/3 of X\Nminus 4 thirds, and if we add 4
Dialogue: 0,0:26:03.36,0:26:07.100,Default,,0000,0000,0000,,to both sides, we can write this\Nas Y is 2X over 3.
Dialogue: 0,0:26:08.63,0:26:12.56,Default,,0000,0000,0000,,And we've got minus 4 thirds\Nhere already, and we're bringing
Dialogue: 0,0:26:12.56,0:26:17.20,Default,,0000,0000,0000,,over four will be adding four.\NSo finally will have two X over
Dialogue: 0,0:26:17.20,0:26:23.27,Default,,0000,0000,0000,,3. And four is the same as 12\Nthirds, 12 thirds. Subtract 4
Dialogue: 0,0:26:23.27,0:26:28.03,Default,,0000,0000,0000,,thirds is 8 thirds. So that's\Nthe equation of the line.
Dialogue: 0,0:26:28.61,0:26:34.23,Default,,0000,0000,0000,,I want to do that same\Nargument when were given two
Dialogue: 0,0:26:34.23,0:26:37.81,Default,,0000,0000,0000,,arbitrary points instead of\Ntwo specific points.
Dialogue: 0,0:26:39.03,0:26:43.88,Default,,0000,0000,0000,,So suppose we have the point a\Ncoordinates X one and Y 1.
Dialogue: 0,0:26:44.84,0:26:48.28,Default,,0000,0000,0000,,And be with coordinates X2Y2\Nand. Let's see if we can figure
Dialogue: 0,0:26:48.28,0:26:51.44,Default,,0000,0000,0000,,out what the equation of the\Nline is joining these two
Dialogue: 0,0:26:51.44,0:26:55.46,Default,,0000,0000,0000,,points, and I think in this case\Na graph is going to help us.
Dialogue: 0,0:26:56.24,0:26:58.28,Default,,0000,0000,0000,,Just have a quick sketch.
Dialogue: 0,0:26:59.04,0:27:05.07,Default,,0000,0000,0000,,So I've got\Na point A.
Dialogue: 0,0:27:05.76,0:27:08.47,Default,,0000,0000,0000,,Coordinates X One Y1.
Dialogue: 0,0:27:09.41,0:27:13.69,Default,,0000,0000,0000,,And another point B\Ncoordinates X2Y2 and were
Dialogue: 0,0:27:13.69,0:27:19.04,Default,,0000,0000,0000,,interested in the equation of\Nthis line which is joining
Dialogue: 0,0:27:19.04,0:27:19.58,Default,,0000,0000,0000,,them.
Dialogue: 0,0:27:21.07,0:27:26.01,Default,,0000,0000,0000,,Suppose we pick an arbitrary\Npoint on the line anywhere along
Dialogue: 0,0:27:26.01,0:27:30.50,Default,,0000,0000,0000,,the line at all, and let's call\Nthat point P.
Dialogue: 0,0:27:31.09,0:27:34.57,Default,,0000,0000,0000,,P is an arbitrary point, and\Nlet's suppose it's coordinates
Dialogue: 0,0:27:34.57,0:27:37.70,Default,,0000,0000,0000,,are X&Y. For any arbitrary X&Y\Non the line.
Dialogue: 0,0:27:38.61,0:27:43.94,Default,,0000,0000,0000,,And what we do know is that the\Ngradient of AP is the same as
Dialogue: 0,0:27:43.94,0:27:45.71,Default,,0000,0000,0000,,the gradient of a bee.
Dialogue: 0,0:27:46.37,0:27:49.67,Default,,0000,0000,0000,,Let me write that\Ndown the gradient.
Dialogue: 0,0:27:50.73,0:27:57.16,Default,,0000,0000,0000,,Of AP. Is\Nequal to the gradient.
Dialogue: 0,0:27:57.31,0:28:01.19,Default,,0000,0000,0000,,Of a B.
Dialogue: 0,0:28:01.19,0:28:04.85,Default,,0000,0000,0000,,Let's see what that means. Well,\Nthe gradient of AP.
Dialogue: 0,0:28:06.40,0:28:08.33,Default,,0000,0000,0000,,Is just the difference in the Y
Dialogue: 0,0:28:08.33,0:28:12.84,Default,,0000,0000,0000,,coordinates. Is Y minus\NY one over the
Dialogue: 0,0:28:12.84,0:28:16.62,Default,,0000,0000,0000,,difference in the X\Ncoordinates, which is X
Dialogue: 0,0:28:16.62,0:28:18.04,Default,,0000,0000,0000,,minus X one?
Dialogue: 0,0:28:19.13,0:28:23.12,Default,,0000,0000,0000,,So that's the gradient of this\Nline segment between amp and
Dialogue: 0,0:28:23.12,0:28:27.48,Default,,0000,0000,0000,,that's got to equal the gradient\Nof the line segment between A&B.
Dialogue: 0,0:28:28.06,0:28:31.72,Default,,0000,0000,0000,,And once the gradient of the\Nline segment between A&B, well,
Dialogue: 0,0:28:31.72,0:28:35.39,Default,,0000,0000,0000,,it's again. It's the difference\Nof the Y coordinates, which now
Dialogue: 0,0:28:35.39,0:28:37.38,Default,,0000,0000,0000,,is Y 2 minus Y 1.
Dialogue: 0,0:28:37.39,0:28:42.90,Default,,0000,0000,0000,,Divided by the difference in the\NX coordinates which is X2 minus
Dialogue: 0,0:28:42.90,0:28:47.49,Default,,0000,0000,0000,,X one. So that is a formula\Nwhich will tell us.
Dialogue: 0,0:28:48.16,0:28:50.87,Default,,0000,0000,0000,,The equation of a line passing\Nthrough two arbitrary points.
Dialogue: 0,0:28:50.87,0:28:54.12,Default,,0000,0000,0000,,Now we don't usually leave it in\Nthat form. It's normally written
Dialogue: 0,0:28:54.12,0:28:57.10,Default,,0000,0000,0000,,in a slightly different form,\Nand it's normally written in a
Dialogue: 0,0:28:57.10,0:29:01.17,Default,,0000,0000,0000,,form so that all the wise appear\Non one side and all the ex is
Dialogue: 0,0:29:01.17,0:29:04.42,Default,,0000,0000,0000,,appear on another side, and we\Ncan do that by dividing both
Dialogue: 0,0:29:04.42,0:29:06.32,Default,,0000,0000,0000,,sides by Y 2 minus Y 1.
Dialogue: 0,0:29:07.01,0:29:14.06,Default,,0000,0000,0000,,And multiplying both sides by X\Nminus X One which moves that up
Dialogue: 0,0:29:14.06,0:29:21.09,Default,,0000,0000,0000,,to here. And that's the form\Nwhich is normally quoted as the
Dialogue: 0,0:29:21.09,0:29:26.40,Default,,0000,0000,0000,,equation of a line passing\Nthrough two arbitrary points.
Dialogue: 0,0:29:28.85,0:29:32.13,Default,,0000,0000,0000,,Let's use that in an example.
Dialogue: 0,0:29:32.69,0:29:38.100,Default,,0000,0000,0000,,Suppose we have\Ntwo points A.
Dialogue: 0,0:29:39.00,0:29:44.35,Default,,0000,0000,0000,,Coordinates one and minus two\Nand B which has coordinates
Dialogue: 0,0:29:44.35,0:29:46.49,Default,,0000,0000,0000,,minus three and not.
Dialogue: 0,0:29:47.22,0:29:52.58,Default,,0000,0000,0000,,Let me write down the formula\Nagain. Why minus why one over Y
Dialogue: 0,0:29:52.58,0:29:58.34,Default,,0000,0000,0000,,2 minus Y one is X Minus X one\Nover X2 minus X one?
Dialogue: 0,0:29:59.01,0:30:02.43,Default,,0000,0000,0000,,With pop everything we know into\Nthe formula and see what we get.
Dialogue: 0,0:30:03.20,0:30:10.34,Default,,0000,0000,0000,,So we want Y minus Y1Y one is\Nthe first of the Y values which
Dialogue: 0,0:30:10.34,0:30:11.77,Default,,0000,0000,0000,,is minus 2.
Dialogue: 0,0:30:12.00,0:30:16.94,Default,,0000,0000,0000,,Why 2 minus? Why one is the\Ndifference of the Y values that
Dialogue: 0,0:30:16.94,0:30:20.40,Default,,0000,0000,0000,,zero? Minus minus 2.
Dialogue: 0,0:30:20.91,0:30:27.13,Default,,0000,0000,0000,,Equals X minus X one is the\Nfirst of the X values, which is
Dialogue: 0,0:30:27.13,0:30:32.90,Default,,0000,0000,0000,,one and X2 minus X. One is the\Ndifference of the X values.
Dialogue: 0,0:30:32.90,0:30:35.12,Default,,0000,0000,0000,,That's minus three, subtract 1.
Dialogue: 0,0:30:35.98,0:30:41.16,Default,,0000,0000,0000,,And just to tidy this up on the\Ntop line here will get Y +2 on
Dialogue: 0,0:30:41.16,0:30:43.11,Default,,0000,0000,0000,,the bottom line. Here will get
Dialogue: 0,0:30:43.11,0:30:47.32,Default,,0000,0000,0000,,+2. X minus one there on\Nthe right at the top and
Dialogue: 0,0:30:47.32,0:30:49.40,Default,,0000,0000,0000,,minus 3 - 1 is minus 4.
Dialogue: 0,0:30:51.87,0:30:56.11,Default,,0000,0000,0000,,Again, we can tie this up a\Nlittle bit more to will go into
Dialogue: 0,0:30:56.11,0:30:57.63,Default,,0000,0000,0000,,minus 4 - 2 times.
Dialogue: 0,0:30:58.19,0:31:02.37,Default,,0000,0000,0000,,And if we multiply everything\Nthrough by minus, two will get
Dialogue: 0,0:31:02.37,0:31:07.69,Default,,0000,0000,0000,,minus two Y minus four equals X\Nminus one, and we can write this
Dialogue: 0,0:31:07.69,0:31:12.25,Default,,0000,0000,0000,,in lots of different ways. For\Nexample, we could write this as
Dialogue: 0,0:31:12.25,0:31:14.15,Default,,0000,0000,0000,,minus two Y minus X.
Dialogue: 0,0:31:15.05,0:31:18.69,Default,,0000,0000,0000,,And we could add 1 to both\Nsides to give minus 3 zero.
Dialogue: 0,0:31:18.69,0:31:21.21,Default,,0000,0000,0000,,That's one way we could leave\Nthe final answer.
Dialogue: 0,0:31:22.43,0:31:26.25,Default,,0000,0000,0000,,Another way we could leave it as\Nwe could rearrange it to get Y
Dialogue: 0,0:31:26.25,0:31:29.53,Default,,0000,0000,0000,,equals something. So if I do\Nthat, I'll have minus two Y
Dialogue: 0,0:31:29.53,0:31:30.89,Default,,0000,0000,0000,,equals X and if we.
Dialogue: 0,0:31:31.67,0:31:37.09,Default,,0000,0000,0000,,Add 4 to both sides will get\Nplus three there, and if we
Dialogue: 0,0:31:37.09,0:31:41.68,Default,,0000,0000,0000,,divide everything by minus two\Nwill get minus 1/2 X minus
Dialogue: 0,0:31:41.68,0:31:45.85,Default,,0000,0000,0000,,three over 2. So all of these\Nforms are equivalent.
Dialogue: 0,0:31:47.48,0:31:53.85,Default,,0000,0000,0000,,Now, that's not quite the whole\Nstory. The most general form of
Dialogue: 0,0:31:53.85,0:31:58.10,Default,,0000,0000,0000,,equation of a straight line\Nlooks like this.
Dialogue: 0,0:31:58.43,0:32:03.51,Default,,0000,0000,0000,,And earlier on in this unit,\Nwe've seen some equations
Dialogue: 0,0:32:03.51,0:32:09.10,Default,,0000,0000,0000,,written in this form. Let's look\Nat some specific cases. Suppose
Dialogue: 0,0:32:09.10,0:32:14.18,Default,,0000,0000,0000,,that a this number here turns\Nout to be 0.
Dialogue: 0,0:32:14.74,0:32:16.47,Default,,0000,0000,0000,,What will that mean if a is 0?
Dialogue: 0,0:32:17.00,0:32:20.18,Default,,0000,0000,0000,,But if a is zero, we can\Nrearrange this and write BY.
Dialogue: 0,0:32:20.92,0:32:22.93,Default,,0000,0000,0000,,Equals minus C.
Dialogue: 0,0:32:24.01,0:32:27.52,Default,,0000,0000,0000,,Why is minus C Overby?
Dialogue: 0,0:32:29.27,0:32:32.94,Default,,0000,0000,0000,,And what does this mean?\NRemember the A and the beat and
Dialogue: 0,0:32:32.94,0:32:36.61,Default,,0000,0000,0000,,the CIA just numbers their\Nconstants. So when a is zero, we
Dialogue: 0,0:32:36.61,0:32:40.59,Default,,0000,0000,0000,,find that this number on the\Nright here minus C over B is
Dialogue: 0,0:32:40.59,0:32:44.88,Default,,0000,0000,0000,,just a constant. So what this is\Nsaying is that Y is a constant.
Dialogue: 0,0:32:45.72,0:32:48.43,Default,,0000,0000,0000,,Now align where why is constant.
Dialogue: 0,0:32:49.00,0:32:53.81,Default,,0000,0000,0000,,Must be. A horizontal line,\Nbecause why doesn't change
Dialogue: 0,0:32:53.81,0:32:57.23,Default,,0000,0000,0000,,the value of Y is always\Nminus C Overby.
Dialogue: 0,0:32:59.08,0:33:04.22,Default,,0000,0000,0000,,So if you have an equation of\Nthis form where a is zero that
Dialogue: 0,0:33:04.22,0:33:05.32,Default,,0000,0000,0000,,represents horizontal lines.
Dialogue: 0,0:33:06.53,0:33:09.58,Default,,0000,0000,0000,,What about if be with zero?
Dialogue: 0,0:33:09.91,0:33:15.37,Default,,0000,0000,0000,,We're putting B is 0 in here,\Nwill get the AX Plus Co.
Dialogue: 0,0:33:16.06,0:33:20.98,Default,,0000,0000,0000,,And if we rearrange, this will\Nget AX equals minus C and
Dialogue: 0,0:33:20.98,0:33:24.67,Default,,0000,0000,0000,,dividing through by AX is minus\NC over A.
Dialogue: 0,0:33:25.99,0:33:30.57,Default,,0000,0000,0000,,Again, a encia constants so this\Ntime what this is saying is that
Dialogue: 0,0:33:30.57,0:33:31.97,Default,,0000,0000,0000,,X is a constant.
Dialogue: 0,0:33:32.74,0:33:35.58,Default,,0000,0000,0000,,Now lines were X is a constant.
Dialogue: 0,0:33:36.24,0:33:40.96,Default,,0000,0000,0000,,Must look like this. They are\Nvertical lines because the X
Dialogue: 0,0:33:40.96,0:33:42.25,Default,,0000,0000,0000,,value doesn't change.
Dialogue: 0,0:33:42.25,0:33:46.36,Default,,0000,0000,0000,,So this general case\Nincludes both vertical lines
Dialogue: 0,0:33:46.36,0:33:47.90,Default,,0000,0000,0000,,and horizontal lines.
Dialogue: 0,0:33:49.03,0:33:52.66,Default,,0000,0000,0000,,So remember, the most general\Nform will appear like that.
Dialogue: 0,0:33:54.20,0:33:57.13,Default,,0000,0000,0000,,Provided that be isn't zero,\Nyou can always write the
Dialogue: 0,0:33:57.13,0:34:00.65,Default,,0000,0000,0000,,equation in the more familiar\Nform Y equals MX plus C, but
Dialogue: 0,0:34:00.65,0:34:04.46,Default,,0000,0000,0000,,in the case in which B is 0,\Nyou get this specific case
Dialogue: 0,0:34:04.46,0:34:05.92,Default,,0000,0000,0000,,where you've got vertical\Nlines.