Video Transcript
A line in the 𝑥𝑦-plane has a
slope of negative two-thirds and passes through the point 12, three. What is the 𝑦-intercept of this
line?
To figure out what the 𝑦-intercept
is for this line, we must first understand where the 𝑦-intercept is located on the
equation of a line. So the equation of a line is 𝑦
equals 𝑚𝑥 plus 𝑏, where 𝑚 is the slope and 𝑏 is the 𝑦-intercept, where it
crosses the 𝑦-axis. So for this line, we will be
looking for 𝑏.
So in this question, we’re given
two things. We’re given the slope, also known
as 𝑚, is negative two-thirds. And it passes through the point 12,
three. So there’s two routes we could take
to solve this question.
Looking at the equation of a line,
we have 𝑦 equals 𝑚𝑥 plus 𝑏. So we could plug in negative
two-thirds for 𝑚 and then 𝑥 and 𝑦, 12 and three. And then all that would be left for
us to find would be 𝑏, the 𝑦-intercept.
The other route would be using the
point slope formula. When you’re given a point that a
line goes through and its slope, there’s a formula that we can use to find the exact
equation of the line. And once we have the equation of
the line, we could find the 𝑦-intercept.
Let’s use this first method and
then go through the second method using the point slope formula. So plugging in these values, we
will have three equals negative two-thirds times 12 plus 𝑏. And we can solve for 𝑏. So we need to take negative
two-thirds times 12. Well, three goes into 12 four
times. And then we can take four times
negative two to be negative eight. And then to solve for 𝑏, we take
three plus eight to give us 11.
So the 𝑦-intercept is where this
line crosses the 𝑦-axis. So if we were to draw this line on
the 𝑥𝑦-plane, we would cross the 𝑦-axis at 11, which would be here. And this is the point zero, 11. So the 𝑦-intercept would be the
point zero, 11.
Now just take a closer look at this
graph. Once we’ve plotted the
𝑦-intercept, we used the slope to get the rest of the points that are on the
line. And the slope is negative
two-thirds. So it’s the rise over the run, so
the vertical change over the horizontal change. So if we have negative two-thirds
from this point at zero, 11, we should go down two and right three. So we will be around here.
Two points is enough to make a
line. However, let’s do one more, which
will be somewhere around here. We will actually need this sixth
spot on the 𝑥-axis. But here we would have our
line.
So now let’s solve for the
𝑦-intercept using the other method, the point slope formula. 𝑦 minus 𝑦 one equals 𝑚 times 𝑥
minus 𝑥 one, where 𝑚 is the slope, so negative two-thirds, and 𝑥 one, 𝑦 one is
the point that the line goes through, so 12, three. So plugging these values in, we
will simplify and then we will have the equation of the line. So we should get 𝑦 equals negative
two-thirds 𝑥 plus 11, because we found the 𝑦-intercept to be 11.
So first, let’s distribute the
negative two-thirds. So we take negative two-thirds
times 𝑥, which is negative two-thirds 𝑥, and negative two-thirds times negative
12. Three goes into 12 four times. So negative two times negative four
— be careful not to miss that negative — would give us positive eight.
And now our last step to get it in
the form of 𝑦 equals 𝑚𝑥 plus 𝑏, the equation of a line, we add three to both
sides of the equation. And just as we said, we get 𝑦
equals negative two-thirds 𝑥 plus 11, where negative two-thirds is the slope and 11
is the 𝑦-intercept.
So once again, to answer our
question, the 𝑦-intercept of this line would be the point zero, 11.
