WEBVTT

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A linear function is a function
of the form F of X equals

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a X Plus B where A&B represent

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real numbers. And when we show
this graphically, a represents

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the gradients of the function
and B represents the Y axis

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intersect, which is sometimes
called the vertical intercept.

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Now what do you think would
happen if we varied a? Well,

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let's have a look at a few

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examples. Because we're looking
at the graphs of linear

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functions, that means we're
going to be looking at straight

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lines, and so plot a straight
line. We only need two points,

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however, we often choose three
points because the Third Point

00:00:49.170 --> 00:00:54.406
is a good check to make sure we
haven't made a mistake, so let's

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have a look at F of X equals X

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+2. OK, first points I look
at is F of 0.

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Now F of zero 0 + 2,
which is simply too.

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S is one.

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Is 1 + 2, which gives

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us 3. An F of two.

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2 + 2 which will give us

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4. OK, for the next function,
let's look at F of X equals

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2 X +2.

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F of X equals 2 X +2.

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So we get F of 0.

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Equals 2 * 0, which is 0 + 2,
which gives us 2.

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S is one which gives us 2 *
1 which is 2 + 2, which gives

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us 4. And F of two which gives
us 2 * 2, which is 4 + 2,

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which gives us 6.

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There's no reason why I
shouldn't be negative, so let's

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look a few negative values. If
we had F of X equals minus two

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X +2. We would have FO

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equals. Minus 2 * 0 which is
0 + 2, which gives us 2.

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F of one which gives us minus 2

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* 1. Which is minus 2 +
2, which gives us 0.

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An F of two which gives us minus
2 * 2 which is minus 4 +

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2 which gives us minus two. And
finally we'll look at F of X.

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Equals minus X

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+2. So we've got
F of 0.

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Equals 0 + 2, which is 2.

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S is one which equals minus 1
+ 2, which equals 1. And finally

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F of two which is minus 2
+ 2 which equals 0.

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Now what we're interested in
doing is looking at the graphs

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of these functions. So if we
have our axes drawn with F of X

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on the vertical scale an X on
the horizontal axis, the first

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function we looked at was F of X
equals X +2, which gave us

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points at 02.

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Second point resort.

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13 Our third
points was at 2 full.

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And when we join this up, we
expect a straight line.

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We can

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label less.
F of X.

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Equals X +2.

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The second function we looked up
was F of X equals 2 X +2.

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Which games, the points 02,
which we've already marked here,

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was the .1 four.

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And it gave us the .2.

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6.

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We should be able to draw these
with a straight line.

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We can label
SF of X.

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Equals 2 X +2.

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The next function we looked up
was F of X equals minus two X

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+2, and once again this gave us
a points at 02 appoint at one

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zero and a point at two and

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minus 2. And when we join
these up as before, we

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expect a straight line.

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We can label less.

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F of X.

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Equals minus two X +2 and the
final function we looked at was

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F of X equals minus X +2
and this gave us a point at

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02 again point at one one.

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Anna points AT20.

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We can join those up to get
a straight line.

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This is
F of

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X. Equals minus
X plus so.

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Now first thing we notice about
these graphs is that they all

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crossed 2 on the F of X axis.
That's be'cause be value is 2 in

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every single function and be
represents the Y axis intercept.

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What we were interested in is
what happens as the value of a

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changes. Now when A is positive,
the line goes up and the bigger

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the value of A, the faster the
line goes up as X increases.

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And when A is negative, the line

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goes down. And the bigger the
value of an absolute terms, the

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faster the line goes down as X

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increases. OK, So what happens
as we very be?

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Well, that's always good place
to start is by actually looking

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at few examples. So let's
consider the example F of X

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equals 2X plus 3.

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F of 0 here would be 2
* 0 + 3, which is 0

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+ 3, which is just three.

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F of one is 2 * 1, which gives

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Me 2. Plus three, which gives

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me 5. An F of

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two. Gives Me 2 * 2 which is 4.

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Plus three, which gives me 7.

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OK, Next One next functional
look at is F of X

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equals 2X plus one.

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OK, for this function we get F
of 0 is equal to 2 * 0, which

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is 0 plus one, which gives me

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one. I have one gives Me 2 * 1
which is 2 plus one which gives

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me 3. And F of two gives
Me 2 * 2, which is 4 +

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1, which gives me 5.

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And the final function I want to
look at is F of X equals

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2X minus three.

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F of X equals 2X
minus three, so F of

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0. Is 2 times here, which is
zero takeaway 3 which is minus

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3. F of one.

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2 * 1 which is 2 takeaway. Three
gives me minus one and finally F

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of two. Which is 2 * 2,
which is 4 takeaway three, which

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gives me one.

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So what we're interested in
doing is looking at the graphs

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of these functions.

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So as usual, we have RF of
X on the vertical axis and

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X one horizontal axis. So
first function we talked

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about was F of X equals 2X
plus three and the points

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we had were zero and three.

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15

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And two.

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And Seven. We can join

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those up. With a
straight line label

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up F of
X equals 2X

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plus 3. The next
function we looked up was F of X

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equals 2X plus one and the
points we had there were.

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Zero and one.

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One and three.

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Two and five.

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Once again, we can draw those.

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Join those up with a ruler.

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Label at
one F

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of X.
Equals 2X plus one.

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And the final function looked up
was F of X equals 2X minus

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three. And the points we had
were 0 - 3.

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One and minus one.

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And two. And warm.

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But enjoying those off.

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As before.

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With a ruler. We label list we
get F of X equals 2X minus

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three. OK, first thing we notice
here is that all the graphs are

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parallel. In fact they have the
same gradients, and that's

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because in each case the value
of a was two. So all the graphs

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have a gradient of two and we
also notice that as we varied B,

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when B was three.

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The graph of the function went
through three on the F of X

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axis. Would be was one the graph
of the function went through one

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on the F of X axis and when be
was minus three. The graph of

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the function went through minus
three on the F of X axis.

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OK, so we know what happens when
I'm being positive and when A&B

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are negative. What happens if
A&BRO? Well, let's see what

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think about what happens when a
equals 0 first of all.

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So if A equals 0 we get a
function of the form F of X

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equals a constant, so that could
be for example, F of X equals 2.

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Or F of X equals minus three.
Just a couple of examples.

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We can sketch what they might

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look like. F of X axis here.

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Now X axis here F of X equals 2.
That means for.

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Whatever the value of X,
the F of X values always

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two. So in fact we just
get a horizontal line.

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Which comes through two on the F
of X axis.

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So if of X equals 2.

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And when F of X equals minus
three, we get a horizontal line.

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That just comes through.

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Minus three on F of X axis.

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So that's what happens when a
equals 0. What about when B

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equals 0? But let's have a look.

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The B equals 0. We get a
function of the form F of X

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equals a X and as we said at the
beginning, a can be any real

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number. So, for example,
we might have F of X

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equals 2X or F of X
equals minus 3X.

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OK, and as we've already said,
what happens when we use?

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These values of AF

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of X&X. For looking at F of
X equals 2X. It's going to come

00:13:38.464 --> 00:13:42.650
through the origin because B
equals 0, so it will cross F of

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X at 0. And it will have a
gradient of two since a IS2.

00:13:50.300 --> 00:13:55.748
So it's a sketch. This could
represent F of X equals 2X.

00:13:56.520 --> 00:14:01.434
Of X equals minus three X once
again will go through the origin

00:14:01.434 --> 00:14:02.946
because B equals 0.

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And it has a gradients of minus

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three. Remember the minus means
the line is coming down and the

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three means that it's going to
be a bit steeper than it was

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before, so it might be like

00:14:13.980 --> 00:14:20.510
this. F of X
equals minus 3X.

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OK, Lastly I want to look at
functions which are not in the

00:14:26.497 --> 00:14:29.638
form F of X equals a X plus B.

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So. What would we do? So we want
our functions in form F of X

00:14:36.348 --> 00:14:40.212
equals X plus B. It's quite
useful, so you can think about

00:14:40.212 --> 00:14:43.110
now if we used Y equals F of X

00:14:43.110 --> 00:14:45.970
just for convenience. So suppose

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I had. 4X minus three

00:14:49.930 --> 00:14:53.990
Y. Equals
2.

00:14:55.060 --> 00:15:01.014
First thing we want to do is
make Y the subject of this

00:15:01.014 --> 00:15:07.426
equation. So if I had three Y
answer both sides 4X equals 2 +

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3 Y. Now I want to get three
wide by itself, so I need to

00:15:13.549 --> 00:15:15.607
take away 2 from both sides.

00:15:16.470 --> 00:15:20.922
So over here I got 4X
takeaway 2 on this side. If

00:15:20.922 --> 00:15:25.003
I take away too, we just get
left with three Y.

00:15:26.360 --> 00:15:31.053
And so finally to make why the
subject I need to divide both

00:15:31.053 --> 00:15:32.136
sides by three.

00:15:32.750 --> 00:15:35.228
So we get 4 thirds of X.

00:15:35.740 --> 00:15:38.770
Minus 2/3 equals

00:15:38.770 --> 00:15:46.271
Y. And as we said before,
Y equals F of X. So this means

00:15:46.271 --> 00:15:52.212
our function is actually F of X
equals 4 thirds X minus 2/3.

00:15:53.100 --> 00:15:58.490
So this function represents a
straight line with the gradients

00:15:58.490 --> 00:16:03.880
of Four Thirds and Y axis
intercept of minus 2/3.

00:16:05.770 --> 00:16:13.582
What about if we had two
X minus 8 Y plus eight

00:16:13.582 --> 00:16:16.837
Y minus one equals 0?

00:16:17.660 --> 00:16:22.616
Once again, we want to make why
the subject of the equation so a

00:16:22.616 --> 00:16:26.510
natural first step would be to
add 1 to both sides.

00:16:27.100 --> 00:16:32.630
So 2X plus eight Y
equals 1.

00:16:34.680 --> 00:16:40.215
Next thing you want to do to get
8. Why by itself is to subtract

00:16:40.215 --> 00:16:44.643
2 X from both sides. If we
subtract 2 actually miss side,

00:16:44.643 --> 00:16:49.809
we just get left with a Y and
this side we get one takeaway

00:16:49.809 --> 00:16:54.517
2X. And finally we need to
divide both sides by eight since

00:16:54.517 --> 00:16:56.419
we just want why we've got

00:16:56.419 --> 00:17:02.236
eight, why there? So divide
both sides by it. We got Y

00:17:02.236 --> 00:17:07.924
equals 1/8 - 2 over 8 times X
and obviously ones are

00:17:07.924 --> 00:17:13.612
functioning to form a X Plus B,
which means we would change

00:17:13.612 --> 00:17:18.826
around. Just rearrange this
right son side here to get Y

00:17:18.826 --> 00:17:22.618
equals minus 2 eighths of X
plus 1/8.

00:17:23.710 --> 00:17:28.704
And we can simplify minus 2
eighths to be minus 1/4.

00:17:29.230 --> 00:17:36.490
So we get minus one quarter of X
Plus one 8th. And as we said

00:17:36.490 --> 00:17:39.394
before, Y equals F of X.

00:17:39.530 --> 00:17:40.550
So here we have it.

00:17:41.100 --> 00:17:46.938
We are function is F of X equals
minus one quarter X Plus one

00:17:46.938 --> 00:17:51.108
8th, and graphically this is
represented by a straight line

00:17:51.108 --> 00:17:55.695
with the gradients of minus 1/4
and yx intercept of 1/8.

00:17:56.240 --> 00:18:01.448
What about if we
have this example?

00:18:02.600 --> 00:18:08.768
Y equals. 13
X minus 8.

00:18:09.280 --> 00:18:11.060
All divided by 5.

00:18:12.160 --> 00:18:13.840
Now a little why is already the

00:18:13.840 --> 00:18:17.648
subject of the formula. It's not
quite in the required form, and

00:18:17.648 --> 00:18:19.394
that's because of this divide by

00:18:19.394 --> 00:18:26.798
5. But we can just rewrite the
right hand side as Y equals 13 X

00:18:26.798 --> 00:18:33.950
divided by 5 - 8 / 5 and
since why is F of X? We can

00:18:33.950 --> 00:18:37.526
write this as F of X equals 13

00:18:37.526 --> 00:18:41.296
over 5X. Minus 8 over

00:18:41.296 --> 00:18:46.186
5. So this function is
represented graphically by a

00:18:46.186 --> 00:18:51.477
straight line with the gradients
of 13 over 5 and a Y axis

00:18:51.477 --> 00:18:53.512
intercept of minus 8 fifths.