Video Transcript
Consider the graph. Which of the following could be the
equation of the line?
So within this question, weโre
given a graph and weโre given five possibilities for what the equation of this line
could be. Now, the key word in this question
is โcould.โ Itโs not saying this is the
equation of the line; itโs just saying which is a realistic possibility.
From looking at the graph, we donโt
have any scales on either the ๐ฅ- or the ๐ฆ-axis. So it wouldnโt be possible to
determine the equation of the line exactly. Instead, weโre just considering
which of these lines do or donโt have the right characteristics to be the equation
of this line.
Each of these lines are in
slope-intercept form: ๐ฆ equals ๐๐ฅ plus ๐. Weโll answer this question by
considering whether each line has a slope and ๐ฆ-intercept that could match up with
those in the diagram.
Letโs look at the slope of the line
in the diagram first of all. Itโs sloping upwards from left to
right, which means it is a positive slope. Therefore, the value of ๐ for this
line must be positive. Letโs look at the ๐ฆ-intercept,
which is this point here. That point is below the
๐ฅ-axis. And therefore, the ๐ฆ-intercept
must be negative. So we know that in the equation of
this line ๐ must be greater than zero, but ๐ must be less than zero.
Now, letโs look at the five
possibilities that weโre given and weโll look at the signs of both ๐ and ๐.
In the first equation, both ๐ and
๐ are positive, which means this canโt possibly be the equation of the line that
weโre looking for. For ๐ฆ equals negative a third ๐ฅ
plus two, ๐ is negative and ๐ is positive. But that doesnโt match up with what
weโre looking for. So we have to rule this one out as
well. For the line ๐ฆ equals a third ๐ฅ
minus two, the slope ๐ is positive; itโs one-third and the ๐ฆ-intercept ๐ is
negative; itโs negative two. And this does match up with what
weโre looking for. So this equation is a possibility
for the line.
For the equation ๐ฆ equals negative
a third ๐ฅ minus two, both ๐ and ๐ are negative. So this equation does not match up
with the line. Finally, for ๐ฆ equals one-third ๐ฅ
plus two, both ๐ and ๐ are positive this time. And again, this doesnโt match up
with the information in the diagram: ๐ should be negative. So this equation is ruled out.
Hence, only one of the five
equations has a slope and a ๐ฆ-intercept that could plausibly be the slope and the
๐ฆ-intercept of the line in the diagram based on their signs.
Therefore, our answer to the
problem is that the only equation which could be the equation of the line is ๐ฆ
equals one-third ๐ฅ minus two.
