### Video Transcript

Determine the equation of the line
which passes through point negative seven, negative eight and itโs perpendicular to
the line meeting the ๐ฅ-axis at nine and the ๐ฆ-axis at one.

The first word weโre gonna look at
is perpendicular cause weโre told that the line that we need to find the equation of
and the other line are perpendicular to one another. But how does that help us? Well it helps us because actually
if we have the slope of two lines that perpendicular to each other, then if you
multiply their slopes together you get an answer of negative one. So we can actually use that to find
the slope of one line if we know the slope of the other.

Okay, great! So now weโll need to find the slope
of one of our lines. Well first of all, weโve actually
got two points that we can write down from the information weโve got, cause we know
that the line meets the ๐ฅ-axis at nine and the ๐ฆ-axis at one. So therefore, we know that there
are two points, the first point being nine, zero. And thatโs cause it meets the
๐ฅ-axis at nine. So that means the ๐ฅ-coordinate is
gonna be nine. before I actually cross into the ๐ฅ-axis, then the ๐ฆ is gonna be
zero. And if we meet the ๐ฆ-axis at one,
then it means the ๐ฅ-coordinate at that point is gonna be zero. So itโs gonna be zero, one.

Okay, great! So weโve got two points, but how
are these helpful? Well, we actually have a formula to
help us find out the slope. And that is that ๐, our slope, is
equal to ๐ฆ two minus ๐ฆ one over ๐ฅ two minus ๐ฅ one. And what this actually means is the
change in ๐ฆ divided by the change in ๐ฅ. So when we have two points, we see
the change in ๐ฆ between those points and then we divided by the change in ๐ฅ
between those points, okay? So letโs do that using the two
points we have.

So what Iโve done is Iโve actually
labeled our coordinates to help us. So weโve got ๐ฅ one, ๐ฆ one and ๐ฅ
two, ๐ฆ two. And then what Iโm gonna do is
actually call our slope ๐ one. So to find ๐ one, weโre gonna now
use the formula with the values that weโve got. Therefore, we can say that ๐ one
is gonna be equal to one minus zero, cause thatโs our ๐ฆ two minus our ๐ฆ one, over
zero minus nine, our ๐ฅ two minus our ๐ฅ one, which should give us a slope of
negative one over nine.

Okay, great! So weโve now found the slope of the
other line. So what we want to see now is find
the slope of the line that we want to find the equation of. And we can do that using our rule
that we know for perpendicular lines cause we know that ๐ one multiplied by ๐ two
is gonna be equal to negative one. So therefore, negative one over
nine multiplied by ๐ two is equal to negative one. So what we want to is actually
divide both sides of the equation here by negative one over nine. So when we do that, we get ๐
two. So our slope of the line that we
are looking for is equal to negative one over negative one over nine.

Alright, so how did we work that
out? What does it mean? Well actually what it means is the
negative reciprocal. So we can say that actually itโs
gonna be the negative reciprocal of negative one over nine. So therefore, ๐ two is gonna be
equal to nine. Well we get that because actually
dividing by negative one over nine is the same as multiplying by negative nine over
one. Well if you multiply negative one
by negative nine over one, you just get nine cause negative multiplied by a negative
is a positive.

Okay, great! So weโve now found ๐ two. Well now what we want to do is
actually determine the equation of the line. And we have a point that it goes
through, which is negative seven and negative eight, and we have its slope, which is
equal to nine. So therefore, we can actually do is
use the slope-point equation, which is a general equation, to help us find the
equation of the straight line. And this tell us that ๐ฆ minus ๐ฆ
one is equal to ๐ multiplied by ๐ฅ minus ๐ฅ one, where ๐ฆ one and ๐ฅ one are the
coordinates of our point.

Okay, so letโs use this to actually
find the equation of our line. So what weโve have done is actually
substituted in our ๐ฅ- and ๐ฆ- values. So we have ๐ฆ minus, and then weโve
got negative eight thatโs because our ๐ฆ coordinate was negative eight, then is
equal to nine because thatโs our slope multiplied by ๐ฅ minus. Then weโve got negative seven
because that was our ๐ฅ-coordinate. Well then, itโs worth nothing at
this point weโve actually got ๐ฆ minus negative eight and ๐ฅ minus negative
seven. And if you actually subtract the
negative then it turns into an add, so actually ๐ฆ add eight. And weโve got ๐ฅ add seven.

So next what Iโm gonna do is
actually expand the parentheses. So when I do that, I get ๐ฆ plus
eight is equal to nine ๐ฅ cause nine multiplied by ๐ฅ gives us nine ๐ฅ and then add
63, and thatโs because nine multiplied by positive seven gives us positive 63. So now what Iโm gonna do is
actually to subtract eight from each side of the equation. And when we do that, we get ๐ฆ is
equal to nine ๐ฅ plus 55.

So this is one way that we can
leave the equation. So we can leave it in the form of
๐ฆ equal ๐๐ฅ plus ๐. And the other ๐ฆ we can actually
leave it is as nine ๐ฅ minus ๐ฅ plus 55 is equal to zero. So we leave it equal zero we do
that by subtract ๐ฆ from each side of the equation. Okay, great! So weโve now solved the problem,
and we said that the equation of the line which passes through point negative seven
negative eight and is perpendicular to the line meets the ๐ฅ-axis at nine and the
๐ฆ-axis at one is ๐ฆ equals nine ๐ฅ plus 55 or nine ๐ฅ minus ๐ฆ plus 55 equal
zero.