Video Transcript
Find the equation of the line
with 𝑥-intercept three and 𝑦-intercept seven and calculate the area of the
triangle on this line and the two coordinate axes.
So this question has two
parts. We’re first asked to find the
equation of a line and then we’re asked to calculate the area of this
triangle. I think a diagram would be
helpful here in order to visualize the situation. So we have a pair of coordinate
axes. We’re told that this line has
𝑥-intercept three, which means it cuts the 𝑥-axis at three. We’re also told the line has
𝑦-intercept seven, so it cuts the 𝑦-axis at seven. By connecting these two points,
I have the line that I’m looking to find the equation of and I can see the
triangle that I’m asked to find the area of. It’s this triangle here.
So let’s start with the first
part of this question, which asks to find the equation of this line. I’m going to do this using the
slope-intercept form, 𝑦 equals 𝑚𝑥 plus 𝑐. I can work out one of these two
values straight away. Remember 𝑐 represents the
𝑦-intercept of the line. And I’m told in the question
that this is equal to seven. So the equation of the line is
𝑦 equals 𝑚𝑥 plus seven. I now need to work out the
slope of this line. And in order to do so, I need
the coordinates of two points on the line. Well, I can use the coordinates
of these points, the 𝑥-intercept and the 𝑦-intercept.
The slope of the line remember
is calculated as the change in 𝑦 divided by the change in 𝑥. So looking at my diagram and
using these two points, I’m gonna find the change in 𝑦 first of all. I can see that as I move from
left to right across the diagram, the 𝑦-coordinate changes from seven to zero,
which is a change of negative seven. It’s really important that you
consider this change in 𝑦 as negative seven, not seven. The line is sloping downwards
from left to right, and therefore it has a negative gradient. Now, let’s look at the change
in 𝑥. I can see that as I move from
left to right across this diagram, the 𝑥-coordinate changes from zero to three,
which gives me a change in 𝑥 of positive three.
Now, I can substitute the
change in 𝑦 and the change in 𝑥 into my calculation for the slope of this
line. And we have that the slope of
the line is equal to negative seven over three. Finally, in order to complete
the first part of the question and find the equation of the line, I need to
substitute this value for 𝑚 into the equation. I have then that the equation
of this line is 𝑦 equals negative seven over three 𝑥 plus seven. Now, sometimes you may be asked
to give your answer in a slightly different format, for example, a format that
doesn’t involve fractions. So you’d need to multiply the
equation there by three, but as it hasn’t been specified here I’m going to leave
my answer as it is now. So that’s the first part of the
question completed.
The second part asked me to
calculate the area of the triangle formed by this line and the two coordinate
axes. Now from the diagram, we can
see that this is a right-angled triangle because the 𝑥- and 𝑦-axes meet at a
right angle. To find the area of a
right-angled triangle, we need to multiply the base by the perpendicular height
and then divide by two. So looking at the diagram, I
can see that the base of this triangle is that measurement of three units. The height of the triangle is
seven units. Now, we refer to this as
negative seven when we’re calculating the slope of the line because the
direction was important. For when we’re just looking at
the length of that line in order to calculate an area, we’ll take its positive
value of seven. So our calculation for the area
is three multiplied by seven divided by two. And this gives us an answer of
10.5 square units for the area of this triangle.
