### Video Transcript

A straight line ๐ฟ has the equation
๐ฆ equals three ๐ฅ minus two. Find the equation of the line
perpendicular to ๐ฟ that passes through the point four, four.

Okay, when we try to solve this
problem and we want to find the equation of this new line, the keyword is
perpendicular, as this gives us an idea of the relationship between our straight
line ๐ฟ and this new line. The way this helps us is because
actually it talks about the slopes between the two lines. Because if theyโre perpendicular, it
means the product to the slopes is equal to negative one, but how that can help us
find out the slope of our new line?

If we look at another definition of
the relationship of the two lines, we can see that the slope of perpendicular lines
are the negative reciprocal of one another. And actually, we can use this
trying to work out what the slope will be of our new line. Well, looking at our original
equation, weโve got ๐ฆ equals three ๐ฅ minus two. Well, we know that the coefficient
of ๐ฅ is going to be the slope of ๐ฟ. So that means we can say the slope
of ๐ฟ is three and we know that because of ๐ฆ equals ๐๐ฅ plus ๐, where ๐ is the
slope and ๐ is the ๐ฆ-intercept.

Great! So we now got the slope of ๐ฟ. Weโre gonna find the slope of our
new line and weโre gonna do that using our definition for the relationships between
the slopes of perpendicular lines and the original line.

First of all, we know itโs negative
because itโs saying itโs the negative reciprocal of one another. So in this case, because it was
positive three is the slope of ๐ฟ, then the slope of ๐, our perpendicular, is going
to be negative.

And then the second part is gonna
be one-third. And the reason itโs one-third is
because of this word here reciprocal. And reciprocal what itโs actually
means is what you multiply a number by to get one. For example, three multiplied by
third gives you three-thirds, which equals to one.

Great! And we can check that weโve got it
right as well by using the first definition we looked at: the product of the slopes
needs to be equal to negative one. Well, three multiplied by negative
a third is equal to negative one. So yes, great! Weโve now found the slope of our
new line.

We can now move on to work out the
equation of the line, and weโre gonna do that using this. And this is the point-slope
equation, where ๐ is the slope, which we discussed before, and ๐ and ๐ are the
๐ฅ- and ๐ฆ-coordinates of a point that is on that line. So we can now start to sub these
values in. So ๐, so we got ๐ฆ minus ๐. ๐ is the ๐ฆ-coordinate; well,
weโve got this point over here, which is four, four. So our ๐ฆ-coordinate is four.

So we have ๐ฆ minus four is equal
to our slope, which is negative a third multiplied by and then inside the
parentheses, we have ๐ฅ minus ๐; well, ๐ is our ๐ฅ-coordinate, which is also four,
which gives us ๐ฆ minus four is equal to negative a third ๐ฅ minus four.

Great! So now all we need to do is to
simplify this to have our final equation. Okay, to simplify it to the next
stage, weโll expand the parentheses to let ๐ฆ minus four is equal to negative a
third ๐ฅ. And then be careful of this bit;
weโve got negative a third multiplied by negative four, which will give us positive
four over three.

Fab! Weโre almost there, just one more
stage. And to do that, we need to add four
to both sides of the equation. To enable us to do that more
easily, Iโd actually convert this four into thirds, so that would give us 12
thirds. So now, we can calculate our final
equation, which is ๐ฆ is equal to negative a third ๐ฅ plus 16 over three; itโs 16
over three as you got four over three plus 12 over three gives us 16 over three. Fantastic! So weโve now managed to find the
equation of the line perpendicular to ๐ฟ that passes through the point four,
four.

So just a quick recap to see how
weโve done that. So first of all, if you find an
equation of another line, check to see if itโs parallel. If itโs parallel, it will have the
same slope. If itโs perpendicular, it will have
the negative reciprocal of the slope. Once youโve got that, you use the
point-slope equation and substitute the value of a point that you have, and then
youโll be able to find your new equation.