### Video Transcript

Find the equation of the line
πΏ. Give your answer in the form π¦
equals ππ₯ plus π.

So π¦ equals ππ₯ plus π is just a
general form of the equation for a straight line. The letter π represents a number,
which is the gradient of the line. And the letter π represents a
number, which is the π¦-intercept of the line. So letβs just recall what those
things mean.

The gradient, well one definition
of the gradient is by how much does the π¦-coordinate change when I increase the
π₯-coordinate by one. After this, we could just pick a
point on the line, increase the π₯-coordinate by one, and then find our way back to
the line.

And in finding our way back to the
line, what happens to the corresponding π¦-coordinate? Well, in this case, it goes down
from nine to seven. And thatβs a decrease of two or
negative two. So in this case, the gradient is
negative two.

So what is the π¦-intercept? Well, we can interpret that as
meaning what is the π¦-coordinate when the line cuts the π¦-axis. Now in this case, hereβs the point
where the line cuts the π¦-axis. And thatβs the point zero,
five. So the π₯-coordinate is zero. But the π¦-coordinate is five. So in our case, π is equal to
five.

Now we know the value of π and the
value of π, the value of the gradient, and the value of the π¦-intercept. We can write out our equation. So the answer is π¦ equals negative
two π₯ plus five.

Now in this question, weβve been
given a nice graph that had a nice integer gradient of negative two. Now itβs not quite as simple as
that in all cases. So just before we go, Iβm gonna go
through another method of working out the gradient. So an alternative definition for
gradient is that the gradient is the difference in π¦-coordinates divided by the
difference in π₯-coordinates between two points. And to use this method as
accurately as possible, what you need to do is pick two points on the line,
preferably as far away as possible, but both with preferably integer π₯- and
π¦-coordinates. So thatβs π₯- and π¦-coordinates
which are whole numbers.

Now letβs think about it. In getting from the first point to
the second point, my π₯-coordinate has increased by five and my π¦-coordinate has
decreased by 10. So putting those numbers into our
little formula, the difference in π¦-coordinates is negative 10. And the difference in
π₯-coordinates is positive five. And negative 10 divided by positive
five is negative two. So Iβve come up with the same
gradient, negative two.

Now that sounds all well and
good. But it seems a lot more complicated
than the first method we used. So why would we ever choose to use
it? Well, sometimes the gradients we
get are not nice integers. For example, with this green line,
if I move from my first point to my second point, again the π₯-coordinate increases
by five. But this time the π¦-coordinate
decreases by eight. So my gradient will be negative
eight divided by positive five.

Now that wouldnβt have been so easy
to calculate accurately using the original first method that I gave you. If I increase my π₯-coordinate by
just one, in this case, the π¦-coordinate decreases by one and a bit. Is that 1.5? Is it 1.6? Is it 1.7? Itβs very difficult to tell. But if Iβve picked nice integer
values for my π₯- and π¦-coordinates here, it makes it much easier to calculate an
accurate gradient. So going back to our original
question then, the answer is π¦ equals negative two π₯ plus five.