Video Transcript
In this video, we will learn how to
represent a straight line in the two-intercept form using the 𝑥- and
𝑦-intercepts. We should already be familiar with
some of the other forms in which the equation of the straight line can be given. For example, we have the
slope–intercept form, 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑚 represents the slope of the
line and 𝑏 represents the 𝑦-intercept. We also have the point–slope form,
𝑦 minus 𝑦 one equals 𝑚 𝑥 minus 𝑥 one, where 𝑚 is the slope of the line as
before and 𝑥 one, 𝑦 one are the coordinates of any point that lies on the
line.
The various forms of the equation
of a straight line are useful in different contexts because they reveal different
features of the graph of a straight line. For example, it is very easy to
determine the slope and 𝑦-intercept of a straight line given its equation in
slope–intercept form. The different forms also allow us
to find the equation of the straight line when we’re given different sets of
information.
The two-intercept form of the
equation of a straight line uses the fact that many straight lines intercept each of
the 𝑥- and 𝑦-axes exactly once. There are, however, two exceptions
to this, horizontal and vertical lines, which are each parallel to one of the
coordinate axes and so don’t intercept it. We can’t therefore write the
equations of horizontal and vertical lines in the two-intercept form. However, all diagonal lines do
intercept each coordinate axis exactly once. We’re going to label these points
as 𝑎, zero for the point where the line intercepts the 𝑥-axis and zero, 𝑏 for the
point where it intercepts the 𝑦-axis. The values of 𝑎 and 𝑏 are the 𝑥-
and 𝑦-intercepts of the line, respectively.
We then define the two-intercept
form of the equation of a straight line as follows. The two-intercept form of the
equation of the straight line that intercepts the 𝑥-axis at 𝑎, zero and intercepts
the 𝑦-axis at zero, 𝑏 is 𝑥 over 𝑎 plus 𝑦 over 𝑏 is equal to one. The derivation of this form of the
equation of a straight line is actually relatively straightforward. We know that, in general, the slope
of the line passing through the points with coordinates 𝑥 one, 𝑦 one and 𝑥 two,
𝑦 two is given by 𝑚 equals 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one, that is,
the change in 𝑦 over the change in 𝑥. If we substitute the coordinates of
the 𝑥- and 𝑦-intercepts for our line, so those are the coordinates 𝑎, zero and
zero, 𝑏, then we have 𝑚 equals zero minus 𝑏 over 𝑎 minus zero, which simplifies
to negative 𝑏 over 𝑎.
Next, we recall the point–slope
form of the equation of a straight line, 𝑦 minus 𝑦 one equals 𝑚 𝑥 minus 𝑥 one,
where 𝑚 is the slope and 𝑥 one, 𝑦 one are the coordinates of any point on the
line. Substituting the slope we’ve just
calculated and the point 𝑎, zero for 𝑥 one, 𝑦 one gives 𝑦 minus zero equals
negative 𝑏 over 𝑎 multiplied by 𝑥 minus 𝑎. We can distribute the parentheses
on the right-hand side to give 𝑦 equals negative 𝑏 over 𝑎 𝑥 plus 𝑏. Dividing both sides of the equation
by 𝑏 gives 𝑦 over 𝑏 equals negative 𝑥 over 𝑎 plus one. And finally, adding 𝑥 over 𝑎 to
each side of the equation gives 𝑥 over 𝑎 plus 𝑦 over 𝑏 equals one. And we’ve now arrived at the
two-intercept form of the equation of the straight line which intercepts the 𝑥-axis
at 𝑎, zero and the 𝑦-axis at zero, 𝑏.
We should notice, of course, that
𝑎 and 𝑏, the 𝑥- and 𝑦-intercepts, are the denominators of the two quotients. And so we can immediately identify
the 𝑥- and 𝑦-intercepts of a straight line given its equation in this form. We could also work the other way,
and we can write down the equation of the straight line in the two-intercept form if
we’re given its 𝑥- and 𝑦-intercepts. We will now consider some examples
in which we use the two-intercept form of the equation of a straight line. In our first example, we’ll
practice finding the equation of a straight line in this form when given the
coordinates of the points at which it intercepts each axis.
If a straight line intercepts the
𝑥-axis at six, zero and intercepts the 𝑦-axis at zero, five, write the equation of
the straight line in two-intercept form.
Let’s recall first the
two-intercept form of the equation of a straight line. The two-intercept form of the
equation of the straight line that intercepts the 𝑥-axis at 𝑎, zero and intercepts
the 𝑦-axis at zero, 𝑏 is 𝑥 over 𝑎 plus 𝑦 over 𝑏 equals one. We’re told that this line
intercepts the 𝑥-axis at six, zero, so the value of 𝑎 is six. We’re also told that the line
intercepts the 𝑦-axis at zero, five, so the value of 𝑏 is five. Substituting 𝑎 equals six and 𝑏
equals five into the two-intercept form of the equation of a straight line gives our
answer. 𝑥 over six plus 𝑦 over five is
equal to one.
Let’s now consider an example which
is essentially the reverse of this. We’ll see how to identify the 𝑥-
and 𝑦-intercepts of a line given its equation in two-intercept form.
List the coordinates of the
𝑥-intercept and the 𝑦-intercept of the line 𝑥 over three minus 𝑦 over two is
equal to one.
Now, looking carefully at the form
in which the equation of this line has been given, we should notice that it looks
very similar to the two-intercept form of the equation of a straight line. The two-intercept form is 𝑥 over
𝑎 plus 𝑦 over 𝑏 equals one, where the line intercepts the 𝑥-axis at 𝑎, zero and
intercepts the 𝑦-axis at zero, 𝑏. However, if we look really closely
at this equation, we see that there is a subtraction sign rather than an addition
sign between the two terms on the left-hand side. We therefore need to manipulate the
equation we’ve been given so that it matches up fully with the two-intercept
form. And we’ll then be able to use it to
determine the coordinates of the 𝑥- and 𝑦-intercepts.
From our knowledge of algebra, we
know that subtracting 𝑦 over two is the same as adding negative 𝑦 over two or we
can think of this as adding 𝑦 over negative two. It doesn’t matter whether we write
that negative in the numerator or denominator of the quotient. So we can take this equation 𝑥
over three minus 𝑦 over two equals one and rewrite it as 𝑥 over three plus 𝑦 over
negative two equals one. The only change is in that second
term. Now we wouldn’t usually choose to
leave a negative value in the denominator of a quotient like this. But by doing this, our equation now
matches up exactly with the format of the two-intercept form of the equation of a
straight line.
We can therefore determine the 𝑥-
and 𝑦-intercepts by considering the denominators of the two quotients. The denominator for the 𝑥-term is
three, which tells us that the value of 𝑎 is three, and so the coordinates of the
𝑥-intercept are three, zero. Looking at the second term, we see
that the denominator for 𝑦 is negative two. This tells us then that the value
of 𝑏 is negative two, and so the coordinates of the 𝑦-intercept are zero, negative
two. By manipulating the equation we
were given slightly so that it perfectly matches up with the two-intercept form of
the equation of a straight line, we found that the coordinates of the 𝑥-intercept
of this line are three, zero and the coordinates of the 𝑦-intercept are zero,
negative two.
Now, in this example, we needed to
be quite careful because the 𝑦-intercept had a negative value. In the same way, we also need to be
careful if either of the intercepts has a fractional value.
Let’s consider the straight line
with equation five 𝑥 plus seven 𝑦 is equal to one. At first glance, we may think that
the equation of this line is in two-intercept form with 𝑎 equal to five and 𝑏
equal to seven. We may then conclude that the line
intercepts the 𝑥-axis at the point five, zero and the 𝑦-axis at the point zero,
seven, but this would be incorrect. If we compare this equation
carefully with the two-intercept form of the equation of a straight line, we see
that the 𝑥- and 𝑦-terms should be divided by the constants which represent the 𝑥-
and 𝑦-intercepts, not multiplied by them.
The trick we need to realize is
that five 𝑥 is equivalent to 𝑥 divided by one-fifth. This is because 𝑥 multiplied by
five is the same as 𝑥 divided by one over five, which we can write as 𝑥 over
one-fifth. In the same way, seven 𝑦 is
equivalent to 𝑦 over one-seventh. So this equation can be rewritten
as 𝑥 over one-fifth plus 𝑦 over one-seventh is equal to one, and this equation is
now in two-intercept form. We can now correctly conclude that
the value of 𝑎 is one-fifth, so the line intercepts the 𝑥-axis at the point
one-fifth, zero. And the value of 𝑏 is one-seventh,
so the line intercepts the 𝑦-axis at the point zero, one-seventh.
Sometimes the equations we’re given
will be in other forms, such as the point–slope or slope–intercept form of the
equation of a straight line. Converting between the various
forms is an essential skill because the different forms are useful for identifying
different properties of the line. Let’s now look at an example in
which we’ll rearrange the equation of a straight line given in slope–intercept form
into the two-intercept form.
Write the equation of the line 𝑦
equals negative two 𝑥 plus six in the two-intercept form.
We recall first that the
two-intercept form of the equation of a straight line, which intercepts the 𝑥-axis
at 𝑎, zero and intercepts the 𝑦-axis at zero, 𝑏, is 𝑥 over 𝑎 plus 𝑦 over 𝑏 is
equal to one. We therefore need to take the
equation we’ve been given and rearrange it. We begin by adding two 𝑥 to each
side of the equation, which gives two 𝑥 plus 𝑦 is equal to six. We’ve now collected the 𝑥- and
𝑦-terms on one side of the equation, with the constant term on the other. But this equation isn’t in
two-intercept form because the constant term on the right-hand side must be equal to
one. We therefore need to divide both
sides of the equation by six. Doing so gives two 𝑥 over six plus
𝑦 over six is equal to one.
We can then simplify the first
quotient by canceling a factor of two in both the numerator and denominator to give
𝑥 over three plus 𝑦 over six equals one. And this equation is now in the
two-intercept form. Although we haven’t specifically
been asked to do this, we can use this form to determine the 𝑥- and 𝑦-intercepts
of this straight line. The value of 𝑎 is three, and the
value of 𝑏 is six.
Now as we saw in this example, it’s
really important that we don’t confuse the two-intercept form of the equation of a
straight line with the general form 𝑎𝑥 plus 𝑏𝑦 equals 𝑐. At this stage here, our equation
was in the general form, but we needed to divide every term in the equation by six
before it was in the two-intercept form.
Let’s now consider how we can find
the slope of a straight line given in the two-intercept form. When we derived the two-intercept
form earlier on, we actually calculated the slope of such a line. We saw that the slope of the line
which passes through the points 𝑎, zero and zero, 𝑏 is negative 𝑏 over 𝑎. We can therefore state this as a
general result. The slope of a straight line given
in two-intercept form 𝑥 over 𝑎 plus 𝑦 over 𝑏 equals one, where 𝑎, zero are the
coordinates of the 𝑥-intercept and zero, 𝑏 are the coordinates of the
𝑦-intercept, is negative 𝑏 over 𝑎. This is a really useful general
result because it enables us to find the slope of a straight line given in the
two-intercept form without needing to rearrange its equation or plot its graph.
Let’s now consider one final
example in which we’ll calculate both intercepts and the slope of a straight line
given in two-intercept form.
The graph of the equation 𝑥 over
four plus 𝑦 over 12 equals one is a straight line. What are the coordinates of the
𝑥-intercept of the line? What are the coordinates of the
𝑦-intercept of the line? What is the slope of the line?
We should notice that the equation
we’ve been given is in the two-intercept form of the equation of a straight line, 𝑥
over 𝑎 plus 𝑦 over 𝑏 equals one. And we know that for a line given
in this form, the coordinates of its 𝑥-intercept are 𝑎, zero and the coordinates
of its 𝑦-intercept are zero, 𝑏. We can therefore determine the
coordinates of the 𝑥- and 𝑦-intercepts of this line by comparing the equation
we’ve been given with the general form and determining the values of 𝑎 and 𝑏,
which are the denominators of the two quotients. We see first that the value of 𝑎,
that’s the value by which 𝑥 is divided, is four, and so the coordinates of the
𝑥-intercept are four, zero. In the same way, the value of 𝑏 is
12, and so the coordinates of the 𝑦-intercept are zero, 12.
Next, we consider the slope of this
line. Now, we could calculate it using
the coordinates of the two points we’ve determined to lie on the line. Or we could recall a general
result, which is that the slope of a straight line whose equation is given in the
two-intercept form is equal to negative 𝑏 over 𝑎. Our value of 𝑏 we’ve just
determined to be 12 and the value of 𝑎 is four. So we have 𝑚 equals negative 12
over four, which is equal to negative three. So we’ve completed the
question. The coordinates of the 𝑥-intercept
are four, zero; the coordinates of the 𝑦-intercept are zero, 12; and the slope of
this line is negative three.
Let’s finish by recapping some of
the key points from this video. The two-intercept form of the
equation of the straight line that intercepts the 𝑥-axis at 𝑎, zero and intercepts
the 𝑦-axis at zero, 𝑏 is 𝑥 over 𝑎 plus 𝑦 over 𝑏 equals one. The values of 𝑎 and 𝑏 give the
𝑥- and 𝑦-intercepts of the straight line. The slope of a straight line whose
equation is given in two-intercept form is equal to negative 𝑏 over 𝑎. We saw also that we can convert
between the different forms of the equation of the straight line by rearranging the
equation. And in particular, we must be
careful to distinguish between the two-intercept form and the general form 𝑎𝑥 plus
𝑏𝑦 equals 𝑐.
