Consider the graph. Which of the following could be the
equation of the line?
So within this question, we’re
given a graph and we’re given five possibilities for what the equation of this line
could be. Now, the key word in this question
is “could.” It’s not saying this is the
equation of the line; it’s just saying which is a realistic possibility.
From looking at the graph, we don’t
have any scales on either the 𝑥- or the 𝑦-axis. So it wouldn’t be possible to
determine the equation of the line exactly. Instead, we’re just considering
which of these lines do or don’t have the right characteristics to be the equation
of this line.
Each of these lines are in
slope-intercept form: 𝑦 equals 𝑚𝑥 plus 𝑐. We’ll answer this question by
considering whether each line has a slope and 𝑦-intercept that could match up with
those in the diagram.
Let’s look at the slope of the line
in the diagram first of all. It’s sloping upwards from left to
right, which means it is a positive slope. Therefore, the value of 𝑚 for this
line must be positive. Let’s look at the 𝑦-intercept,
which is this point here. That point is below the
𝑥-axis. And therefore, the 𝑦-intercept
must be negative. So we know that in the equation of
this line 𝑚 must be greater than zero, but 𝑐 must be less than zero.
Now, let’s look at the five
possibilities that we’re given and we’ll look at the signs of both 𝑚 and 𝑐.
In the first equation, both 𝑚 and
𝑐 are positive, which means this can’t possibly be the equation of the line that
we’re looking for. For 𝑦 equals negative a third 𝑥
plus two, 𝑚 is negative and 𝑐 is positive. But that doesn’t match up with what
we’re looking for. So we have to rule this one out as
well. For the line 𝑦 equals a third 𝑥
minus two, the slope 𝑚 is positive; it’s one-third and the 𝑦-intercept 𝑐 is
negative; it’s negative two. And this does match up with what
we’re looking for. So this equation is a possibility
for the line.
For the equation 𝑦 equals negative
a third 𝑥 minus two, both 𝑚 and 𝑐 are negative. So this equation does not match up
with the line. Finally, for 𝑦 equals one-third 𝑥
plus two, both 𝑚 and 𝑐 are positive this time. And again, this doesn’t match up
with the information in the diagram: 𝑐 should be negative. So this equation is ruled out.
Hence, only one of the five
equations has a slope and a 𝑦-intercept that could plausibly be the slope and the
𝑦-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 𝑦
equals one-third 𝑥 minus two.