### Video Transcript

Write the equation of the line with
slope three over two and ๐ฆ-intercept zero, three in the form ๐๐ฅ plus ๐๐ฆ plus ๐
is equal to zero.

In this question, weโre asked to
find the equation of a straight line. Weโre asked to give our answer in
the form ๐๐ฅ plus ๐๐ฆ plus ๐ is equal to zero. And this type of equation for a
straight line has a special name. Itโs called the general form for
the equation of a straight line. So to find this, letโs see what
information weโre given about our straight line. Weโre told the value of the slope
of our straight line is three over two. And weโre also given the
๐ฆ-intercept of our straight line. Thatโs the point zero, three.

And we know a form of a straight
line which uses both the slope and the ๐ฆ-intercept. Itโs called the slopeโintercept
form of a straight line. Thatโs the equation ๐ฆ is equal to
๐๐ฅ plus ๐, where ๐ is the slope of our straight line and ๐ is the value of the
๐ฆ-intercept. And itโs worth pointing out
something here. Usually, in the slopeโintercept
form, the value of ๐ is written as ๐ or ๐. However, in this case, weโve used
both ๐ and ๐ in the general form of the equation of our line, so weโll just rename
this ๐.

And in fact, we already know the
slope of our straight line is three over two. Weโre told this in the
question. And weโre also told the value of
our ๐ฆ-intercept. This is the point zero, three. So the ๐ฆ-intercept of our point is
going to be equal to three. So in the slopeโintercept form of
our straight line, weโll set ๐ equal to three over two and ๐ equal to three. This gives us the equation ๐ฆ is
equal to three over two ๐ฅ plus three. But remember, weโre asked to give
our answer in the general form of the equation of a straight line. Thatโs the form ๐๐ฅ plus ๐๐ฆ plus
๐ is equal to zero.

So we have a few options. Weโre gonna start by multiplying
our equation through by two. This is not technically
necessary. However, it will simplify our
equation. We need to multiply every term in
this equation by two. Three over two times two is three,
and three times two is six. So we get two ๐ฆ is equal to three
๐ฅ plus six. Now, weโll just subtract two ๐ฆ
from both sides of this equation. And this gives us our final answer
of three ๐ฅ minus two ๐ฆ plus six is equal to zero.

However, there is something worth
pointing out. This is not the only answer we
couldโve given. When we had two ๐ฆ is equal to
three ๐ฅ plus six, we subtracted two ๐ฆ from both sides of the equation. But we couldโve also subtracted
three ๐ฅ and subtracted six from both sides of our equation. This would then give us the
equation negative three ๐ฅ plus two ๐ฆ minus six is equal to zero. And this is in the form ๐๐ฅ plus
๐๐ฆ plus ๐ is equal to zero, and it represents the same straight line.

And in fact, we can see that these
two equations represent the same line because if we multiply either through by
negative one, we get the other equation. In fact, we can multiply either
equation through by any nonzero constant, and weโll get an equation for the same
straight line in the general form. But usually, as a rule of thumb, we
try to give our values of ๐, ๐, and ๐ as integers and we try to make our value of
๐ positive.

But any other solution of this form
is also valid. Therefore, we were able to show the
equation of the straight line with slope three over two and ๐ฆ-intercept zero, three
in the form ๐๐ฅ plus ๐๐ฆ plus ๐ is equal to zero is given by three ๐ฅ minus two
๐ฆ plus six is equal to zero.