1.1 LINES IN THE PLANE
In this section, you will study lines and their equations. A line is a straight line.
Defmition of the Slope of a Line
(There are several notations to describe slope. Some are listed below. We may use any of
them in this class during the course of the year. )
The slope m of a line passing through the points is (xP yJand(x2 , yJ
~
m = y2 - y1
x2 - x 1
= change in y = Ay = rise ~
change in x tJx run x ' 5
A\]'t'Y"O..~
"'ft::t
~
*When using the formula above, remember the order ofsubtraction is important.
EXAMP~d the slope of a line p~ugh the following po~
A. (-2, 5) and (3, -1)
B. (9, 7) and (6, 7)
C. (3,4) and (3, -8)
'-------"'
7-7 _Q, -;:. 0
s-- r
-5
q - 4-
3
Example la): Sketch a graph through the point (2, 3) with the
b) undefmed
c) 2
A) 0
-r
ind~ated
slopes on the same graph.
d) -113
-I
..
-3
If the slope of a line:
1. Has positive slope (m > 0), the line rises from left to right.
2. Has negative slope (m < 0), the line falls from left to right.
3. Has zero slope (m = 0), the line is horizontal.
4. Has undefined slope
(m
-- anumober) , the line is vertical.
39
The following are equations of lines. They are helpful in either sketching graphs of lines or fmding
equations of lines when given a graph.
Equations of Lines
1. y=mx+b
qs'((j) y- y 1 = m(x- x 1)
Point-slope form
3. Ax+ By+ C == 0
General form or Standard form nl)
oj~ x=a uru:l. slo)'LP
S ~ y==b
D
o.nSw-exs iY\
Slope-interceptform
slo~
Vertical line
Horizontal line
6:~Write your answer in slope
EXAMPLE 2 Find an equation of a line given two points (
intercept _fQrm. Use the graphing utility t<? verify.
-
~- ~ 1 ~
0
'r'f"t~.Ci1'd\-\ s.
YY\
l)( -~ ~
- 2.
----...
'
;t-
7
3:..
O
(o - (
-
(
"f. "' -{_ ()( --1)
;JO-=-
-7_)(_
+-lLf]
EXAMPLE 3 Given the following equations, fmd the slope and they -intercept of each and sketch.
A. y=tc
B. ~ -3y=2._ -l"
c. x=-8
D. y=.7x
-J
A . Slope= 0
y-int= JL
B. Slope = '-/3
y-int = -s-/5
unci
C. Slope =
y-int = ....llJ.)r.....
D. Slope= P 7
y-int= Q_
~~J~-~
PARALLEL AND PERPENDICULAR LINES
<(I 1. Parallel lines have same (equal) slopes.
#
40
2. Perpendicular lines hav~posite reciprocal slopes. Eg.
~
=- -
1
~
•17
(
{0
EXAMPLE 4 Find an equation of the line that passesf %tugh the point (2, 1) and is a) parallel
b) perpendicular to ~ - 2 y = 3 ~ Lf X
L!_ -=- ;;l
-3
o..)
,__ -z
,----z..
pr"l/.d
~ ~
-
2
IV\ Lx-
t ::
a
~
-z.
h) pe~~c;Jttr
~ - ?1 i-lk( )( -/0
1 ::
»-1 . "' 2-~
- ~ (x -2.)
'ef- I
d -"i) " -{ x. -1- 1+ J
~=--i_x+-2_
'j. -~ ~ "2 x:- Lf +I
r£-zx-3]
an
EXAMPLE 5 Find
equation of the line that passes through the poin (-1
b) perpendicularto = - ] .
1
6
(c
EXAMPLE 6
Suppose your salary was $28,500 in 1999 and $32,900 in 2001. Assume your salary
~::. "2....
follows a linear growth pattern.
0 rrv~
A. Write a linear equation giving the salary Yin terms of the ~ear x. Let x = 0 be the year 1999.
0 ) '2. 1 ) ~v....J)
., "
a.ve r~ ra.~ oP ~~qll&e
L.(L{Ot) -=-
l....?..C>D
'32~DO -l<i(.S'"ZJb
L-b
~
B. Use the linear equation to predict your salary in 2005.
~ -.;. c., (no 6) +- zy/:::.-D LJ
-z.~
~
d- '8r ::: m
~~
o
,
lX-
1.-oo
tt..
X c)
y -~I ~ u() ::= L "2 () 0
(J
)(
I ~ : :. z-wo -r-
+ 2 ~ s-r::::t::J
r L <t-,
sn o]
~~~t:'f/ 1 700
...
41