Writing Linear Equations in PointSlope Form
Objective Learn how to write equations for
lines given information such as the slope and a point on the
line, or two points on the line.
There are a number of ways to write equations representing
straight lines. This lesson introduces two of them. The
pointslope form can be written if the slope of the line in
question and a point on the line are given, or if two points on
the line are given. The standard form is also useful because it
can be used to represent any line, even a vertical one. Vertical
lines cannot be given in pointslope form since their slope is
not defined.
PointSlope Form
Key Idea
If we have a straight line, and we choose any two different
points on it, we will get the same value for the slope no matter
what points we choose.
Now draw a line like the one shown on the figuer below.
Display some points on the line, such as (1, 2) and (2, 4), in
addition to the origin (0, 0). Then, try to find the slope of
this line (it is actually 2). We write the general point P = ( x
, y ), where x and y are variables. "How can you tell from
the coordinates x and y whether P lies on the line?"
Remember that if (0, 0) and ( x , y ) are used to compute the
slope, the answer still must be 2, since the key idea states that
we will get the same answer no matter which two points are
chosen.
You can find the slope by dividing the rise by the run. Since
the rise is y and the run is x , slope
We know that the slope of this line is 2, so .
So, the rise is twice the run, and y = 2x for any point on
this line. This equation holds for the coordinates of any point
on the line.
There are many equations that describe the same line. We could
have computed the slope using (x, y) and any other point on the
line, such as (1, 2), as shown below.


(y  2) = 2( x  1) 
Multiply each side by x  1. 
Try to write your own equations for the line, using various
points, such as (2, 4) or (3, 6).
Here are some more examples. Determine the slope and write an
equation for each line graphed.
Example 1
Write an equation for the line.
Solution
The line has slope 3. One equation for it is y = 3x .
Example 2
Write an equation for the line.
Solution
The line has slope and an equation for it
is .
So, if the slope of a line (let's call it m) and a single
point (let's call it ( x_{ 0} , y_{ 0} )) on that
line are known, then the following equation for the line can be
written.
( y  y_{ 0} ) = m ( x  x_{ 0} )
This equation is said to be in pointslope form since it uses
only information about one point and the slope.
Now, work on the following excercises.
Exercises
Find an equation in pointslope form for each line.
1. the line with slope 4 going through the point (1, 2)
( y  2) = 4( x  1)
2. the line with slope 1 going through the point (5, 4)
( y  4) = 1( x  5)
3. the line with slope going through the point
(0, 10)
The slope and a single point on the line determine the whole
line. Intuitively, the steepness and one point determine the
line. This can be illustrated by rotating a ruler through a
single point, and tracing out many of the possible slopes.
