James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

appendix B Coordinate Geometry and Lines A13
Example 3 Find an equation of the line through s1, 27d with slope 2 1 2 .
SOLUTIOn Using (3) with m − 2 1 2 , x 1 − 1, and y 1 − 27, we obtain an equation of the
line as
which we can rewrite as
y 1 7 − 2 1 2 sx 2 1d
2y 1 14 − 2x 1 1 or x 1 2y 1 13 − 0 n
Example 4 Find an equation of the line through the points s21, 2d and s3, 24d.
SOLUTIOn By Definition 2 the slope of the line is
m − 24 2 2
3 2 s21d − 2 3 2
Using the point-slope form with x 1 − 21 and y 1 − 2, we obtain
y 2 2 − 2 3 2 sx 1 1d
which simplifies to 3x 1 2y − 1 n
y
b
y=mx+b
Suppose a nonvertical line has slope m and y-intercept b. (See Figure 7.) This means
it intersects the y-axis at the point s0, bd, so the point-slope form of the equation of the
line, with x 1 − 0 and y 1 − b, becomes
y 2 b − msx 2 0d
0
FIGURE 7
x
This simplifies as follows.
4 Slope-Intercept Form of the Equation of a Line An equation of the line
with slope m and y-intercept b is
y − mx 1 b
y
b
x=a
y=b
In particular, if a line is horizontal, its slope is m − 0, so its equation is y − b, where
b is the y-intercept (see Figure 8). A vertical line does not have a slope, but we can write
its equation as x − a, where a is the x-intercept, because the x-coordinate of every point
on the line is a.
Observe that the equation of every line can be written in the form
0
a
x
5
Ax 1 By 1 C − 0
FIGURE 8
because a vertical line has the equation x − a or x 2 a − 0 (A − 1, B − 0, C − 2ad
and a nonvertical line has the equation y − mx 1 b or 2mx 1 y 2 b − 0 (A − 2m,
B − 1, C − 2b). Conversely, if we start with a general first-degree equation, that is, an
equation of the form (5), where A, B, and C are constants and A and B are not both 0, then
we can show that it is the equation of a line. If B − 0, the equation becomes Ax 1 C − 0
or x − 2CyA, which represents a vertical line with x-intercept 2CyA. If B ± 0, the
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.