A line in the 𝑥𝑦-plane has a
slope of eight and passes through the origin. Which of the following is a point
on the line? A) One-eighth, one; B) two,
one-fourth; C) one-fourth, one; or D) one, one-eighth.
So in order to decide which of
these points lies on our line, we first need to find the equation of our line. The equation of a line is 𝑦 equals
𝑚𝑥 plus 𝑏, where 𝑚 is the slope and 𝑏 is the 𝑦-intercept. We are given that the slope is
eight. So we know that the slope is
eight. But what about 𝑏, the
Well, we’re told that this line
passes through the origin. So on an 𝑥𝑦-plane, the origin is
located here, at the point zero, zero. And the 𝑦-intercept is where this
line will cross the 𝑦-axis. Well, here’s the 𝑦-axis. And we know where it’s actually
gonna cross the 𝑦-axis. It’ll cross the 𝑦-axis at zero,
zero. So the value of 𝑦 would be
zero. So we could plug in zero for our
𝑦-intercept. So we would have the equation of a
line as 𝑦 equals eight 𝑥 plus zero, which we don’t have to include the plus
zero. We could just write 𝑦 equals eight
Another way that we could’ve got in
our equation of the line would be we could plug in eight for 𝑚 and then zero for 𝑥
and zero for 𝑦 and then solve for 𝑏, the 𝑦-intercept. So we would have zero equals eight
times zero plus 𝑏. And eight times zero is zero. And then 𝑏 plus zero gives us that
𝑏 is equal to zero, which we’ve already found here.
But as we said in the beginning,
knowing that this line passes through the origin, it’s intuitive to know that the
𝑦-intercept indeed would just be zero. So now let’s go through each of
these points, plug them into our equation, and see if they satisfy the equation. So we need to take each of our
points and plug them into our equation: 𝑦 equals eight 𝑥 plus zero. And again, we can just use 𝑦
equals eight 𝑥. There’s no need to put plus
So, so far, we’ve plugged in all of
the 𝑦-values. And then we put equals eight times
all of the 𝑥-values. And now we’ve done so. So let’s begin with option A). We have one equals eight times
one-eighth. Well, the eights cancel, and we’re
left with one. So one equals one. This satisfies the equation of the
line. So that means the line passes
through this point. But let’s go ahead and check all of
the other options to be sure.
For option B), we have one-fourth
equals eight times two. Well, eight times two is 16. And one-fourth is not equal to
16. So point B), option B), the line
does not pass through this point.
For option C), we get one equals
two because four goes into eight twice, and two times one is two. One is not equal to two. So this is not our answer.
And then, lastly, for option D),
eight times one is eight. And one-eighth is not equal to
eight. So D) is not our answer either.
So as we said before, option A)
will be the correct answer. The point that is actually on this
line would be one-eighth, one.