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

A14
appendix B Coordinate Geometry and Lines
equation can be rewritten by solving for y:
y − 2 A B x 2 C B
y
0 x
(5, 0)
(0, _3)
FIGURE 9
3x-5y=15
and we recognize this as being the slope-intercept form of the equation of a line
(m − 2AyB, b − 2CyB). Therefore an equation of the form (5) is called a linear
equation or the general equation of a line. For brevity, we often refer to “the line
Ax 1 By 1 C − 0” instead of “the line whose equation is Ax 1 By 1 C − 0.”
Example 5 Sketch the graph of the equation 3x 2 5y − 15.
SOLUTIOn Since the equation is linear, its graph is a line. To draw the graph, we can
simply find two points on the line. It’s easiest to find the intercepts. Substituting y − 0
(the equation of the x-axis) in the given equation, we get 3x − 15, so x − 5 is the
x-intercept. Substituting x − 0 in the equation, we see that the y-intercept is 23. This
allows us to sketch the graph as in Figure 9.
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Example 6 Graph the inequality x 1 2y . 5.
SOLUTIOn We are asked to sketch the graph of the set hsx, yd | x 1 2y . 5j and we
begin by solving the inequality for y:
x 1 2y . 5
y
2y . 2x 1 5
2.5
y . 2 1 2 x 1 5 2
y=_ x+
1 2
0
FIGURE 10
5
2
5
x
Compare this inequality with the equation y − 2 1 2 x 1 5 2 , which represents a line with
slope 2 1 2 and y-intercept 5 2 . We see that the given graph consists of points whose
y-coordinates are larger than those on the line y − 2 1 2 x 1 5 2 . Thus the graph is the
region that lies above the line, as illustrated in Figure 10.
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Parallel and Perpendicular Lines
Slopes can be used to show that lines are parallel or perpendicular. The following facts
are proved, for instance, in Precalculus: Mathematics for Calculus, Seventh Edition, by
Stewart, Redlin, and Watson (Belmont, CA, 2016).
6 Parallel and Perpendicular Lines
1. Two nonvertical lines are parallel if and only if they have the same slope.
2. Two lines with slopes m 1 and m 2 are perpendicular if and only if m 1 m 2 − 21;
that is, their slopes are negative reciprocals:
m 2 − 2 1 m 1
Example 7 Find an equation of the line through the point s5, 2d that is parallel to the
line 4x 1 6y 1 5 − 0.
SOLUTIOn The given line can be written in the form
y − 2 2 3 x 2 5 6
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