3.4 The Point-Slope Form of a Line

296 Chapter 3 Linear Functions
Version: Fall 2007
P (−3,2)
y
Q(2,−1)
Figure 3. The line through points
P (−3, 2) and Q(2, −1).
Use the slope formula to determine the slope of the line through the points P (−3, 2)
and Q(2, −1).
We’ll use the point-slope form of the line
m = ∆y −1 − 2
= = −3
∆x 2 − (−3) 5
x
y − y0 = m(x − x0). (7)
Let’s use point P (−3, 2) as the given point (x0, y0). That is, (x0, y0) = (−3, 2). Substitute
m = −3/5, x0 = −3, and y0 = 2 in equation (7), obtaining
y − 2 = − 3
(x − (−3)). (8)
5
This is the equation of the line passing through the points P and Q.
Alternatively, we could also use the point Q(2, −1) as the given point (x0, y0). That
is, (x0, y0) = (2, −1). Substitute m = −3/5, x0 = 2, and y0 = −1 in the point-slope
form (7), obtaining
y − (−1) = − 3
(x − 2). (9)
5
This too, is the equation of the line passing through the points P and Q.
How can the equations (8) and (9) both be the equation of the line through P
and Q, yet look so distinctly different? Let’s place each equation in standard form
Ax + By = C and compare the results.
If we start with equation (8) and distribute the slope,
y − 2 = − 3
(x − (−3))
5
y − 2 = − 3 9
x −
5 5 .
Multiply both sides by the common denominator 5 to clear the fractions.