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.