Page 26 - Calculus Demystified

Basics
CHAPTER 1
y 13

x

Fig. 1.18(b)

1.5 The Equation of a Line

The equation of a line in the plane will describe—in compact form—all the points
that lie on that line. We determine the equation of a given line by writing its
slope in two different ways and then equating them. Some examples best illustrate
the idea.

EXAMPLE 1.10
Determine the equation of the line with slope 3 that passes through the point
(2, 1).

SOLUTION
Let (x, y) be a variable point on the line. Then we can use that variable point
together with (2, 1) to calculate the slope:
y − 1
m = .
x − 2
On the other hand, we are given that the slope is m = 3. We may equate the
two expressions for slope to obtain
y − 1
3 = . (∗)
x − 2
This may be simplified to y = 3x − 5.

Math Note: The form y = 3x − 5 for the equation of a line is called the slope-
intercept form. The slope is 3 and the line passes through (0, 5) (its y-intercept).

Math Note: Equation (∗) may be rewritten as y − 1 = 3(x − 2). In general, the
line with slope m that passes through the point (x 0 ,y 0 ) can be written as y − y 0 =
m(x − x 0 ). This is called the point-slope form of the equation of a line.