Video Transcript
In this video, we will learn how to
find the slope of a line through two points. Let’s begin by looking at the
general equation of a straight line.
The general equation of a straight
line can be written as 𝑦 equals 𝑚𝑥 plus 𝑏. The 𝑚, which is the coefficient of
𝑥, represents the slope or gradient of the line, and it indicates how steep the
line is. The 𝑏-value represents the
𝑦-intercept. Sometimes in the equation of a
straight line, this is indicated by the letter 𝑐 instead. If we visualize a straight line
drawn, the 𝑦-intercept is the point where the line crosses the 𝑦-axis. In this video, however, we’ll just
be focusing on the slope of a line.
Before we begin looking at how we
calculate the slope of a line, there’s a few important things to note. Firstly, this line that we’ve drawn
here would have a positive value for the slope as it slopes upwards from left to
right. This line, however, would have a
negative slope as it slopes downwards from left to right. If we want to compare the slope of
two different lines, particularly if one has a positive slope and one has a negative
slope, then we need to consider the absolute value. Let’s take a look at that.
Let’s say that we have two
lines. The first line has a slope of two,
and the second line has a slope of negative three. We can therefore say that the
absolute value of the slope of line one is two and the absolute value of the slope
of line two is three. Even if we hadn’t drawn these two
lines, we would still be able to say that line two must be steeper since its
absolute value for slope is higher than that of the absolute value for the slope of
the first line. So let’s move on to how we actually
find the slope of a line if we’re given two points on it.
The slope of a line is often
written as the rise over the run. Or alternatively, we can think of
this as the change in 𝑦 divided by the change in 𝑥. There’s also a handy formula that
we can use to calculate the slope of any line between two given points. This formula will work regardless
of whether the slope is positive or negative and regardless of the quadrant of which
our points lie in. Let’s say that we define these two
points as 𝑥 sub one, 𝑦 sub one and 𝑥 sub two, 𝑦 sub two. The slope is then calculated by 𝑦
sub two minus 𝑦 sub one divided by 𝑥 sub two subtract 𝑥 sub one. Remembering that slope is the rise
over the run, then the rise in this case will be calculated by the length of the 𝑦
sub two value subtract the 𝑦 sub one value. The run is calculated by the 𝑥 sub
two value subtract the 𝑥 sub one value.
A handy tip as we go through some
questions involving slope, we must always be careful when we have negative 𝑥- and
𝑦-values. This will mean that sometimes we
need to subtract negative values. But don’t worry if we get a
negative slope in our final answer. That just means that the line would
slope downwards from left to right.
As we go through this video, we’ll
be using this slope formula in each of our questions. So let’s have a look at our first
one.
What is the slope of the line
passing through the points two, two-thirds and six, two?
In order to find this slope or
gradient of the line, we can remember this important formula. The slope of a line is equal to 𝑦
sub two subtract 𝑦 sub one over 𝑥 sub two subtract 𝑥 sub one for two points, 𝑥
sub one, 𝑦 sub one and 𝑥 sub two, 𝑦 sub two.
So let’s take our two points and
define two, two-thirds with the values 𝑥 sub one, 𝑦 sub one and the point six, two
with the values 𝑥 sub two and 𝑦 sub two. We can then plug these into the
slope formula. Our 𝑦 sub two value is two and our
𝑦 sub one value is the fraction two-thirds. On the denominator, we’d have our
𝑥 sub two value of six subtract our 𝑥 sub one value of two.
In order to work out this fraction
calculation on the numerator, we can remember that two is equivalent to
six-thirds. Six-thirds subtract two-thirds
gives us four-thirds. And then on the denominator, we’d
have the value of four since six subtract two is four. In order to work out the answer to
four-thirds over four, we might prefer to think of this in a different way as
four-thirds divided by four. When we’re dividing by a fraction,
that’s equivalent to multiplying by the reciprocal. So we need to calculate four-thirds
multiplied by one-quarter. Taking out a factor of four from
the numerator and denominator before we multiply leaves us with this value of
one-third. The slope of this line would
therefore be one-third.
Before we finish with this
question, let’s check does it matter which point we called 𝑥 sub one, 𝑦 sub one
and which we called 𝑥 sub two, 𝑦 sub two. Let’s switch around our values and
see what happens with the formula. When we fill in our values into the
slope formula, we’d have two-thirds subtract two over two subtract six. We’ll get the same values as
before. Only both the numerator and
denominator will be negative instead of positive. We should remember that when we
multiply two negatives, we get a positive value. So once again, we’d have a slope of
one-third as per our original answer. And we’ve also demonstrated that it
doesn’t matter which points we designate with the value of 𝑥 sub one, 𝑦 sub one
and so on.
In the next question, we’ll need to
find missing values of 𝑥 and 𝑦 given some points on a line and its slope.
Given that the slope of a line
passing through negative two, seven; 𝑥, three; and five, 𝑦 is negative one, find
the values of 𝑥 and 𝑦.
In this question, we’re given three
points. The second point is missing the
𝑥-value and the third point is missing the 𝑦-value. We are however given the slope of
the line joining these three points. In order to answer this question,
we recall that for two points designated with 𝑥 sub one, 𝑦 sub one and 𝑥 sub two,
𝑦 sub two, the slope is equal to 𝑦 sub two subtract 𝑦 sub one over 𝑥 sub two
subtract 𝑥 sub one. So let’s take the point of which we
know the 𝑥- and 𝑦-values, negative two, seven, and designate those with the 𝑥 sub
one, 𝑦 sub one values. The second coordinate 𝑥, three we
can designate as 𝑥 sub two and 𝑦 sub two.
Since we have three points lying on
a line and we’re given its slope, then this will be the same slope between any of
the two points. The slope is given as negative
one. And then our 𝑦 sub two subtract 𝑦
sub one values will be three take away seven. And for our change in 𝑥-values,
we’ll have 𝑥 subtract negative two. On the right-hand side, our
numerator becomes negative four and the denominator is equal to 𝑥 plus two. We can then multiply both sides of
this equation by 𝑥 plus two. Distributing the negative sign
across the parentheses, we have negative 𝑥 subtract two is equal to negative
four. Adding 𝑥 and four to both sides
gives us that two is equal to 𝑥 or alternatively 𝑥 is equal to two. This also means that the second
point could be written as the point two, three.
In order to find the 𝑦-value in
the third coordinate, we’ll use the same slope formula with points one and
three. We can designate point three with
𝑥 sub three and 𝑦 sub three and use these values in the slope formula
accordingly. Plugging in the values and using
the slope of negative one, we have negative one is equal to 𝑦 minus seven over five
subtract negative two. We can then simplify the
denominator to give us a value of seven. We can then multiply both sides of
this equation by seven to give us negative seven is equal to 𝑦 minus seven. Finally, adding seven to both sides
gives us zero equals 𝑦 or rather 𝑦 equals zero. And so this coordinate would be
five, zero.
We’ve now answered this
question. We found that 𝑥 is equal to two
and 𝑦 is equal to zero. But it’s always worth performing
some sort of check on our answers.
One way of doing this would be to
draw a grid, plot these three points, and see if they do indeed lie on a straight
line. We were given that the coordinate
negative two, seven, if 𝑥 is two, then we have the point two, three. And if 𝑦 is zero, then we would
have the point of five, zero. And yes, these would lie on a
straight line. We can also get a nice
demonstration of this slope of negative one. We can see how for every change in
𝑦 unit of one, the change in 𝑥 would also be one. And the line slopes downwards from
left to right, indicating a negative slope. And so we’ve confirmed our two
answers, 𝑥 equals two and 𝑦 equals zero.
In the next question, we’ll find
the slope of a line which is parallel to the 𝑥-axis.
Given that the straight line
passing through the points one, eight and negative six, 𝑘 is parallel to the
𝑥-axis, find the value of 𝑘.
Let’s begin this question by
thinking about what it means for a line to be parallel to the 𝑥-axis. Well, if we draw out our two axes,
the 𝑥- and 𝑦-axis, a line that’s parallel to the 𝑥-axis could look like this or
even like this. Any two lines that are parallel
will have the same slope. If we want to calculate the slope
of a line between two points, we can think of this as the rise over the run or the
change in 𝑦-values over the change in 𝑥-values. So if we take any two points on
this 𝑥-axis, the 𝑥-values on the denominator will change between the two values
that we have chosen, but the 𝑦-values will not change. We can recall that zero divided by
anything will still give us zero. We can therefore say that the
𝑥-axis has a slope of zero, and so does every line that’s also parallel to the
𝑥-axis.
Now that we know that our two
points have a line joining them with a slope of zero, we can use the formula that
the slope is equal to 𝑦 sub two minus 𝑦 sub one over 𝑥 sub two minus 𝑥 sub one
for two points, 𝑥 sub one, 𝑦 sub one and 𝑥 sub two, 𝑦 sub two. We can therefore fill in these
values into our formula, and it doesn’t matter which point we designate with the 𝑥
sub one and 𝑦 sub one values. On the right-hand side then, we’ll
have 𝑘 subtract eight over negative six subtract one. And we recall that the slope of
this line is zero. This simplifies to zero equals 𝑘
minus eight over negative seven. We can multiply both sides of this
equation by negative seven, which gives us zero is equal to 𝑘 minus eight. Adding eight to both sides gives us
eight equals 𝑘 or rather 𝑘 equals eight.
And so, we’ve answered the question
to find the value of 𝑘. But let’s have a quick check. We were given the point one, eight
and another point negative six, 𝑘, which we can’t plot. But we know that the 𝑥-value will
be negative six. The line that joins these two
points is parallel to the 𝑥-axis, which means it’s a horizontal line. This means that the 𝑦-value of
this point must be eight as it’s the coordinate negative six, eight which lies on
this line. And so we’ve shown that our value
of 𝑘 must be equal to eight.
We’ll take a look at one final
question.
What can be said of the slope of a
line that is parallel to the 𝑦-axis?
We can remember that the slope or
gradient of a line indicates how steep a line is. Let’s visualize our 𝑥- and
𝑦-axes. We can even include some lines that
would be parallel to the 𝑦-axis.
The slope of a line between two
points is given as the change in 𝑦-values over the change in 𝑥-values. Alternatively, we sometimes see
this as the rise over the run. So let’s have a think about any two
points on the 𝑦-axis. As the slope is equal to the change
in 𝑦-values over the change in 𝑥-values, we could see that the 𝑦- values would
change, but the 𝑥-values would have no change. The slope is therefore equal to
some value divided by zero. And what do we know about a number
divided by zero? It’s undefined. Anything divided by zero is
undefined. If you’ve ever tried to divide
anything by zero on your calculator, you know that you will get an error
message.
We can then say that the 𝑦-axis
has a slope that is undefined. And therefore, any line which is
parallel will have the same slope. That is, it is undefined.
We’ll now summarize what we’ve
learned in this video. Firstly, we saw that the slope, or
gradient, of a line relates to how steep it is. We saw that there are different
values that are possible for a slope. They can be positive, negative,
zero for horizontal lines, or undefined for vertical lines. Steeper lines will have a higher
absolute value for the slope.
We can find the slope of a line by
thinking of it in terms of the rise over the run or the change in 𝑦 over the change
in 𝑥. Or we can use the formula that for
any two coordinates 𝑥 sub one, 𝑦 sub one and 𝑥 sub two, 𝑦 sub two, the slope is
equal to 𝑦 sub two subtract 𝑦 sub one over 𝑥 sub two subtract 𝑥 sub one,
remembering that when we’re using this formula, it doesn’t matter which point we
designate as 𝑥 sub one, 𝑦 sub one. And when we’re using this formula,
we need to be careful with negative 𝑥- and 𝑦-values.
