### Video Transcript

In this video, we will learn how to
find and write the equation of a straight line in its general form. Weβll see how to do this when weβre
given the coordinates of two points which lie on the line. And weβll also see how we can do
this from the graph of a straight line. You should already be familiar with
how to calculate the slope of a straight line either from two points or from its
graph, although we will review this in our examples.

So the general form of the equation
of a straight line then. The exact letters you use most
often may vary depending on where in the world you are. But the two most commonly used
forms are π¦ equals ππ₯ plus π or sometimes π¦ equals ππ₯ plus π. The values of π and π or π have
important meanings in terms of the graph of the line.

The value π first of all, which is
the coefficient of π₯ in this equation, refers to the slope of the line. For every one unit the line moves
to the right, the line will move π units up. If π is negative, then this will
mean that the line has a negative slope. It slopes downwards from left to
right.

There are lots of different ways
that people think about calculating the slope. The first is as change in π¦ over
change in π₯. Thatβs the vertical change over the
horizontal change between two points that lie on the line. Formally, if those two points have
the coordinates π₯ one, π¦ one and π₯ two, π¦ two, then this can be written as π¦
two minus π¦ one over π₯ two minus π₯ one. The difference in the π¦-values
over the difference in the π₯-values.

Or another more casual way that
people sometimes like to think of this value π is as the rise over the run. Rise being a vertical change and
run being a horizontal change. It doesnβt really matter which way
you prefer to think of it as long as youβre comfortable with calculating the slope
of the line.

Going back to our general form, the
value of π or π refers to the π¦-intercept of the line. Thatβs the π¦-value at which the
line intercepts the π¦-axis. For this reason then, because the
values of π and π or π represent the slope and π¦-intercept of a straight line,
an equation given in this form is sometimes referred to as the slopeβintercept
form.

To find the equation of a straight
line in this general form, we just need to determine the values of these two
constants. Letβs see how we can do this in the
context of an example.

Find the equation of the line that
passes through the points π΄: five, 11 and π΅: 10, 21.

In this question then, weβve been
given the coordinates of two points that lie on the line. Weβre going to find the equation of
this line in its general form. Thatβs π¦ equals ππ₯ plus π or
sometimes π¦ equals ππ₯ plus π, where π represents the slope of the line and π
represents the π¦-intercept.

Weβre going to calculate the slope
or gradient of this line first of all. And we can do this using the
formula change in π¦ over change in π₯ or more formally π¦ two minus π¦ one over π₯
two minus π₯ one, where π₯ one, π¦ one and π₯ two, π¦ two represent the coordinates
of the two points on the line. We can think of point π΄ as π₯ one,
π¦ one and point π΅ as π₯ two, π¦ two, although it doesnβt actually matter which way
round we do this, as long as weβre consistent about the order we subtract for both
π₯ and π¦.

So substituting the values for
these two coordinates, we have 21 minus 11 for the change in π¦ and 10 minus five
for the change in π₯. That gives 10 over five, which is
equal to two.

Now, itβs really important to
remember that we must divide the change in π¦ by the change in π₯. A really common mistake is to do
this the other way round, so to divide the change in π₯ by the change in π¦. But in this case, that would give
five over 10, which is equal to a half. We would get the reciprocal of the
slope we actually need. The other common mistake is to
subtract one set of values in the wrong order. We must make sure that we always
subtract in the same order for both π₯ and π¦. If we donβt, weβll end up with the
exact negative of the slope we want, which again will be incorrect.

Now that we know the slope of the
line is two, we can substitute this value of π into the equation. So we know that the equation of our
line is in the form π¦ equals two π₯ plus π. To determine the value of π, we
use the fact that each of these points, points capital π΄ and capital π΅, lie on our
line, which means we have two pairs of π₯- and π¦-values which must satisfy this
equation.

We donβt need to use both of
them. Letβs just consider point π΄, which
tells us that when π₯ is equal to five, π¦ is equal to 11. We can therefore substitute these
values of π₯ and π¦ into our equation π¦ equals two π₯ plus π to enable us to find
the value of π. Doing so gives 11 equals two
multiplied by five plus π. This is a relatively
straightforward equation to solve. Two multiplied by five is 10. And then if we subtract 10 from
each side, we find that the value of π is one. So we can substitute this value of
π into the equation of our straight line. And we now have that the equation
of this line in its general form is π¦ equals two π₯ plus one.

Now, we use the coordinates of
point π΄ here. But itβs just worth pointing out we
could equally have used the coordinates of point π΅, which tell us that when π₯ is
equal to 10, π¦ is equal to 21. We would then have a different
equation to solve. But it gives the same solution of
π equals one. The equation of the straight line
passing through the two given points then is π¦ equals two π₯ plus one.

Letβs now consider a second example
using a table of values.

Write the equation of the line that
passes through the points indicated in the table of values.

Now, this table is just a different
way of presenting the coordinates of two points that lie on the line. Weβre told that when π₯ is equal to
three, π¦ is equal to 12. So we have the point three, 12. And when π₯ is equal to seven, π¦
is equal to zero. So we also have the point seven,
zero.

We know that the general equation
of a straight line is π¦ equals ππ₯ plus π, where π represents the slope and π
represents the π¦-intercept. To calculate the slope first of
all, we can use the formula change in π¦ over change in π₯. Subtracting the coordinates of our
first point from the coordinates of our second then, we have zero minus 12 for the
change in π¦ and seven minus three for the change in π₯.

Remember, we must make sure we
subtract our coordinates in the same order. We donβt simply subtract the
smaller from the larger. This gives negative 12 over four,
which simplifies to negative three. So the equation of our line is π¦
equals negative three π₯ plus π, and we need to determine the value of π. To do this, we can use the
coordinates of either point as they each lie on the line.

Letβs use the point seven,
zero. We substitute seven for π₯ and zero
for π¦, giving the equation zero equals negative three multiplied by seven plus
π. That simplifies to zero equals
negative 21 plus π. And adding 21 to each side, we find
that π is equal to 21. We can then substitute this value
of π into the equation of our line, and we have our answer. π¦ equals negative three π₯ plus
21.

Sometimes if the values of either
the slope π or π¦-intercept π are fractional, it may be preferable to give our
answer in an equivalent form that doesnβt involve fractions. For example, suppose we found the
equation of a straight line and it was π¦ equals a half π₯ minus four. In order to eliminate the fraction
of one-half, we could multiply the entire equation by the denominator of two. If we do this though, we must
remember to multiply every single term in the equation by two.

So on the left-hand side, weβd have
two π¦. And on the right-hand side, weβd
have π₯ minus eight. A really common mistake would be to
forget to multiply the constant term by two. The equation of this line is
equivalent to π¦ equals a half π₯ minus four, but it doesnβt involve any
fractions.

Now, itβs more usual in this
instance to give our answers in the form ππ₯ plus ππ¦ plus π equals zero, which
just means we group all of the terms on the same side of the equation. And we have zero on the other
side. We could do this by subtracting two
π¦ from each side of the equation to give the equation π₯ minus two π¦ minus eight
is equal to zero. Or equivalently, we could subtract
π₯ and add eight to each side to give the equation negative π₯ plus two π¦ plus
eight equals zero, which is the exact negative of the previous equation we wrote
down.

Another commonly used form is to
group the π₯ and π¦ terms on one side of the equation and the constant term on the
other. So an example of this would be the
equation negative π₯ plus two π¦ equals negative eight. All of these are simply
rearrangements of the original equation, so theyβre all equivalent. We just need to make sure we read
any questions carefully and give our answers in the required form. Letβs look at an example of
this.

Find the equation of the line that
passes through the points π΄: negative 10, two and π΅: zero, five, giving your
answer in the form of ππ¦ plus ππ₯ plus π is equal to zero.

Now, weβve been asked to find the
equation of this straight line in a specific form, with all of the terms on the same
side of the equation. But weβll begin by using the
general form of the equation of a straight line. Thatβs π¦ equals ππ₯ plus π,
where π is the slope of the line and π is its π¦-intercept.

Now, itβs important to be aware
here that this value π is not necessarily the same in our general equation and in
the requested form. Weβre told that the line passes
through the point with coordinates zero, five. So the π¦-intercept of our line is
zero, five. And we can work out straightaway
that the value π in our general form is five.

To work out the value of π, we can
use the formula change in π¦ over change in π₯. Subtracting the coordinates of π΄
from those of π΅, and we have five minus two over zero minus negative 10. We need to be really careful here
because we are subtracting a negative value. A common mistake would be to just
have zero minus 10 in the denominator. But that wouldnβt actually be
subtracting the π₯-coordinate of π΄ from the π₯-coordinate of π΅.

Simplifying, we have three in the
numerator. And in the denominator, zero minus
negative 10 is zero plus 10, which is 10. So the slope of the line is
three-tenths. Substituting the values of π and
π into the equation of our straight line then, and we have π¦ equals three-tenths
of π₯ plus five.

Now, this isnβt in the required
form because weβre asked to collect all the terms on the same side of the
equation. And although the question doesnβt
explicitly say this, the values of π, π, and π that we use should be
integers. We have a fraction of
three-tenths. But we can eliminate this by
multiplying each side by the denominator of 10. Doing so and remembering we need to
multiply every term in the equation, including the constant term, by 10, we get 10π¦
equals three π₯ plus 50. Grouping all the terms on the
left-hand side by subtracting three π₯ and subtracting 50 gives the equation 10π¦
minus three π₯ minus 50 equals zero, which is in the requested form.

Alternatively, we couldβve grouped
the terms on the other side of the equation, which would give the equivalent form
negative 10π¦ plus three π₯ plus 50 equals zero, the exact negative of the equation
weβve given.

So in this example, weβve seen how
to give the equation of a straight line in an alternative but equivalent form. In our next example, weβll see how
we can use knowledge about the π₯- and π¦-intercepts of a straight line to determine
its equation.

What is the equation of the line
with π₯-intercept negative three and π¦-intercept four?

Letβs begin by considering a sketch
of this line. We know its π₯-intercept is
negative three and its π¦-intercept is four. So the line looks a little
something like this. We can see that this line has a
positive slope. So we can use this fact to
sense-check our answer.

First, weβll consider the equation
of this line in its general form π¦ equals ππ₯ plus π. Now, we can actually work out the
value of π straightaway because weβve been given the π¦-intercept explicitly in the
question. The value of π is four. So our equation has the form π¦
equals ππ₯ plus four. To calculate the slope π, we can
use change in π¦ over change in π₯ or the more casual rise over run.

From our sketch, we can see that
the vertical change or the rise is from zero, which will be the π¦-value on the
π₯-axis, to four. Thatβs a change of four units. And the horizontal change is from
negative three to zero. Thatβs a change of three units. That gives a slope of
four-thirds. This value is positive. So this is indeed consistent with
our starting sketch.

Substituting the values of π and
π into the general equation of our straight line then, and we have π¦ equals
four-thirds π₯ plus four. Now, this would be an acceptable
form in which to give our answer, but it does involve fractions. And we often prefer to work without
fractions. So letβs find an equivalent
form.

To eliminate the fractions, we can
multiply the entire equation by the denominator of three. Doing so, and we have three π¦
equals four π₯ plus 12. Remember, we must multiply every
term by three, including the constant term. Now, this would be an acceptable
form in which to give your answer. Or we could choose to collect the
π₯ and π¦ terms on one side and leave the constant term on the other. Subtracting four π₯ from each side
of the equation, and we have our answer in this form. Three π¦ minus four π₯ is equal to
12.

In our final example, weβll
practice finding the equation of a straight line when weβve been given its
graph.

Write the equation represented by
the graph shown. Give your answer in the form π¦
equals ππ₯ plus π.

We recall firstly that, in the
general form of the equation of a straight line π¦ equals ππ₯ plus π, π
represents the slope of the line and π represents the π¦-intercept. We need to determine each of these
values from the graph. We can see the π¦-intercept of this
line straightaway. It crosses the π¦-axis at four. So the value of π is four.

We can calculate the value of π
using the formula change in π¦ over change in π₯. And to do this, we need to identify
any two points on the graph. Weβve already identified the
π¦-intercept. How about we consider the
π₯-intercept? Thatβs the point with coordinates
negative two, zero. From the graph, we can see that
between these two points, the change in π¦ is four units. Itβs a positive change. And the change in π₯ is two units,
again a positive change. So the slope is four over two,
which is equal to two.

Now, remember, the slope of the
line will be the same no matter which points we used to calculate it. We could equally have used the
point negative four, negative four or the point one, six. It doesnβt matter. Weβll get the same slope in each
case. Finally, we substitute the values
of π and π into the general form, giving our answer to the problem. π¦ equals two π₯ plus four.

Letβs now summarize the key points
from this video. Firstly, the general equation of a
straight line is the form π¦ equals ππ₯ plus π or sometimes π¦ equals ππ₯ plus
π, where π represents the slope of the line and π or π represent its
π¦-intercept. For this reason, this form is
sometimes referred to as slopeβintercept form. The slope π can be calculated
either from two points which lie on the line or from a graph using change in π¦ over
change in π₯. This can be formalized as π¦ two
minus π¦ one over π₯ two minus π₯ one. Or we can think of it more casually
as rise over run.

Once weβve found the slope, the
π¦-intercept π or π can be found by substituting the coordinates of any point that
we know lies on the line. Or it can be found from a
graph. To avoid having fractional values,
we can also give the equations of straight lines in alternative equivalent forms,
such as ππ¦ plus ππ₯ plus π equals zero or ππ¦ plus ππ₯ equals π. Although we need to be careful that
weβre aware that the values of π, π, and π will not all be the same between the
three different forms.

When answering questions on finding
the equations of straight lines, we need to make sure we read the question carefully
and always give our answer in the form that has been requested.