Point Slope Form Video Lecture - Mathematics (Maths) Class 11 - Commerce

Hello. I'm Professor Von Schmohawk
and welcome to Why U.
So far we have studied linear equations
of the form "y = mx + b".
This form of the equation of a line
is called the "slope-intercept" form.
The slope-intercept form
involves two constants "m" and "b".
The numerical value of "m"
is the slope of the line.
And the value of "b" determines the point
at which the line crosses the y-axis
called the "y-intercept".
The y-intercept is a point on the y-axis
with coordinates (0,b).
The slope-intercept form
has the advantage of being simple
but is only useful when we know the exact
point where the line crosses the y-axis.
In some cases
when looking at the graph of a line
it may be impossible to determine
the exact coordinates of the point
where the line and y-axis meet.
However, we are often able to discover the
coordinates of some other point on the line.
There is another form of the equation for
a line which we can use in this situation
called the "point-slope" form.
Using the point-slope form,
in addition to the line's slope
we only need to know the coordinates
of one point somewhere on the line.
The point-slope form
of the equation for a line
can be derived by using the formula for slope
which we learned in the previous chapter.
You may recall that this formula uses
the coordinates of any two points on a line
to calculate the line's slope.
If we call the coordinates of these two points
x-one, y-one
and x-two, y-two
then the slope is the difference in the points'
vertical positions, y-two minus y-one
divided by the difference in horizontal positions
x-two minus x-one.
Now, let's say we have a line with a slope "m"
and a point on that line
with coordinates x-one and y-one.
If we use any other point on the line
as the second point in the slope formula
the calculated slope will always be equal to "m".
On the other hand, if we use a point
which does not lie on the line
the calculated slope will not be equal to "m".
So only points which lie on the line
will produce the correct value for "m"
and thus satisfy this formula.
This line can therefore be thought of
as the collection of all points
with coordinates x and y
which satisfy this equation.
This equation is easier to read
if we rearrange it
so that the y's are on the left side
and the x's are on the right.
To do this, we multiply both sides
of the equation by "x minus x-one".
This allows us to cancel the denominator
"x minus x-one" on the left side of the equation.
The result is the point-slope form
of the equation for a line.
Let's try an example, using this form
to write the equation for a specific line.
Let's say that we are given the
graph of a line, but not its equation.
We are told that the line has a slope of three-fifths.
Since we know the slope,
we might try using the slope-intercept form
to write the equation for this graph.
We start by assigning the value of the slope,
three-fifths, to the constant "m".
But from looking at this graph
the exact point where the line crosses
the y-axis is not obvious
so we don't have an exact value
to assign to the constant "b".
But let's say that we know that the line
passes through the point (4,3).
Since we know the slope
and the coordinates of one point
we can write the equation for this line
using the point-slope form.
Since the slope is three-fifths, we assign
a value of three-fifths to the constant "m".
And since the coordinates
of the known point are (4,3)
we set the value of x-one and y-one
to four
and three.
So the equation for this line
in point-slope form
is y minus three
equals three-fifths
times the quantity x minus four.
You may sometimes see the point-slope equation
written with the variable y
alone on the left side.
To write the point-slope equation in this
form, we simply add y-one to both sides
cancelling out the negative y-one on the left
which gives us the point-slope form
with y alone on the left side.
Now let's try another example
using this form.
Once again, we are given the graph of a line
but not its equation.
And once again, it is not clear
from looking at the graph
exactly where the line crosses the y-axis.
We are told that the line passes through
the point (-7,3)
and that the line's slope is negative two-thirds.
Using the point-slope form
we set the value of "m" to the slope
negative two-thirds
and the value of x-one and y-one
to the coordinates of the known point
negative seven
and three.
This equation can be simplified
if instead of subtracting negative seven
we add positive seven.
The equation for this line is now written
in point-slope form.
Since we now have an equation
which describes this line
we can determine the graph's y-intercept.
It is difficult to tell by looking at the graph
exactly where the line intersects the y-axis.
However, we do know that
the x-coordinate of this point must be zero
since every point on the y-axis
has an x-coordinate of zero.
Therefore, if we set the value of x
in our equation to zero
we can solve the equation to find
the corresponding value of y for that point.
Completing the arithmetic
we add zero plus seven
to get seven
and multiply negative two-thirds times seven
which gives us negative fourteen-thirds.
We can then write three as nine-thirds
and add negative fourteen-thirds to nine-thirds
to get negative five-thirds.
So the point on the graph
with an x-coordinate of zero
must have a y-coordinate of negative five-thirds.
The y-intercept is therefore (0, -5/3).
Since we know the y-intercept
if we like, we can now write the equation
for this line in slope-intercept form.
We already know that the slope
is negative two-thirds
so we assign this value to the constant "m".
And since the y-intercept is (0, -5/3)
we set the value of "b" to negative five-thirds.
So the equation for this line
written in slope-intercept form is
"y equals negative two-thirds x, minus five-thirds".
Both of these forms are equivalent ways
to write the equation for the same line.
We can prove this by converting the top equation
to the bottom equation's form.
We do this by distributing negative two-thirds
to the two terms inside the parentheses.
Instead of adding negative two-thirds,
we can simply subtract two-thirds.
We then eliminate the parentheses
and multiply negative two-thirds times seven
to get negative fourteen-thirds.
We can then write three as nine-thirds
and add negative fourteen-thirds to nine-thirds
to get negative five-thirds.
This makes the forms of
these two equations identical.
So although the point-slope and
slope-intercept forms look different
once the proper constant values are assigned
the two equations express the same relation
between x and y and are therefore equivalent.
However, either form may be more convenient
depending upon the information
we know about the line.
In the next lecture, we will introduce an
additional form of the equation for a line
known as the "two-point" form.