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Write the equation in the form ax + b = 0.
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B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * D % ( D ' = -g 6 B f a d e *< 3 < * D D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * D - % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * D - D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * D - T % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * D - T D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * D T s % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * D T s D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * D s % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * D s D ' = % ( D@ ' = % ( D ' =4 @B B B B % ( ) ) ) D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * D % ( D ' = + 4 8 ?R C B B C B # p p t _ w B *Y 3 > B
p p t _ w < * D D ' = + 4 8 ?R C B B C B # p p t _ h B *Y 3 > B
p p t _ h < * D D ' = +K 4 8 ? C B B # p p t _ x + ( c o s ( - 2 * p i * ( 1 - $ ) ) * - # p p t _ x - s i n ( - 2 * p i * ( 1 - $ ) ) * ( 1 - # p p t _ y ) ) * ( 1 - $ ) C B ? B *Y 3 > B
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* % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * <
* D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * >
% ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * >
D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * >
- % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * >
- D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * >
- < % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * >
- < D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * >
< X % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * >
< X D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * >
X h % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * >
X h D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * >
h | % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * >
h | Dr ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * >
| % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * >
| D ' = % ( D+ ' =4 @B B
B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < *
% ( D ' = -g 6 B f a d e *< 3 < *
D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * >
% ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * >
D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < *
% ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < *
D ' = % ( D+ ' =4 @B B
B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < *
% ( D ' = -g 6 B f a d e *< 3 < *
D ' = % ( D= ' =4 @B B 1 B B % ( ) ? ) ? ) D ' = .I 7 B B B B B[ M - 3 . 3 3 3 3 3 E - 6 - 4 . 4 4 4 4 4 E - 6 L 0 . 0 4 5 8 4 0 . 0 9 4 4 5 * 3 >* B
p p t _ x B
p p t _ y =@ 0 B B A A p B B < B lA= < *
D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < *
% ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < *
D ' = % ( DI ' =4 @B B 5 B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < *
% ( D ' = + 4 8 ?R C B B C B # p p t _ w B *Y 3 > B
p p t _ w < *
D ' = + 4 8 ?R C B B C B # p p t _ h B *Y 3 > B
p p t _ h < *
D ' = -g 6 B f a d e *< 3 < *
D ' = % ( DV ' =A @B B 5 B B 0B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < *
% ( D ' = + 4 8 ?R C B B C B # p p t _ w B *Y 3 > B
p p t _ w < *
D ' = + 4 8 ?R C B B C B # p p t _ h B *Y 3 > B
p p t _ h < *
D ' = -g 6 B f a d e *< 3 < *
D ' = % ( D@ ' = % ( D ' =4 @B B B B % ( ) ) ) D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * 0
% ( D ' = + 4 8 ?R C B B C B # p p t _ w B *Y 3 > B
p p t _ w < * 0
D ' = + 4 8 ?R C B B C B # p p t _ h B *Y 3 > B
p p t _ h < * 0
D ' = +K 4 8 ? C B B # p p t _ x + ( c o s ( - 2 * p i * ( 1 - $ ) ) * - # p p t _ x - s i n ( - 2 * p i * ( 1 - $ ) ) * ( 1 - # p p t _ y ) ) * ( 1 - $ ) C B ? B *Y 3 > B
p p t _ x < * 0
D ' = +K 4 8 ? C B B # p p t _ y + ( s i n ( - 2 * p i * ( 1 - $ ) ) * - # p p t _ x + c o s ( - 2 * p i * ( 1 - $ ) ) * ( 1 - # p p t _ y ) ) * ( 1 - $ ) C B ? B *Y 3 > B
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Rewrite as ax + b = 0:
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The solution can be checked algebraically by solving the original equation. Note: The same solution will be found even if you rewrite the equation as 2x + 1 = 0. R 2 M K H
0 h ? 33 U/ M/ _ _ _ P P T 1 0 -/ . _` + nGel DY. ' s =
@B D. ' =
@B A ? %, ( < + O %, (
< + D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * 5 % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * 5 D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * 5 - % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * 5 - D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * 6 % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * 6 D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * 5 / K % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * 5 / K D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * 7 % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * 7 D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * 5 M a % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * 5 M a D
' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * 8 % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * 8 D ' = % ( D+ ' =4 @B B
B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * 9 % ( D ' = -g 6 B f a d e *< 3 < * 9 D ' = % ( D8 ' =A @B B
B B 0B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * : % ( D ' = -g 6 B f a d e *< 3 < * : D ' = % ( D+ ' =4 @B B
B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * ; % ( D ' = -g 6 B f a d e *< 3 < * ; D ' = % ( D ' = % ( D[ ' =4 @B B B B % ( ) ? ) ? D ' = .w 7 B B B B B M 0 . 0 0 0 1 8 - 0 . 0 0 0 9 2 C 0 . 0 0 5 3 8 0 . 0 1 3 2 0 . 0 2 6 3 9 0 . 0 6 9 2 2 0 . 0 3 1 6 0 . 0 8 3 3 4 * 3 >* B
p p t _ x B
p p t _ y =@ 0 B B f f p B B < B ,= < * ; D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * < % ( D ' = -s 6 B w i p e ( d o w n ) *< 3 < * < D ' = % ( D ' = % ( DD ' =A @B B B B 0B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * = % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * = D ' = % ( D+ ' =4 @B B
B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * > % ( D ' = -g 6 B f a d e *< 3 < * > D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * ? % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * ? D ' = % ( D ' = % ( DD ' =A @B B B B 0B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * @ % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * @ + +0 + : 0 + +0 + = 0 + +0 + @ 0 + h 0 ^N0 0 0 z (
0 F Z :
y0 Z : T p `
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X $ APPROXIMATING SOLUTIONS IN REAL LIFE % 2 % @
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A EXAMPLE 4
2
Z
0
C * A j 0 3 1 1 7 8 4 @
F @
0 @
0
0 D( P
; 4.7 ( 2 (
0
0 Д @
] % Solving Linear Equations Using Graphs & 2 & D( H
0 0 ! h ? 33 _ _ _ P P T 1 0 . ]R + D ' =
@B Da ' =
@B A ? %, ( < + O %, (
< + D ' = % ( D@ ' = % ( D ' =4 @B B B B % ( ) ) ) D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * 0 % ( D ' = + 4 8 ?R C B B C B # p p t _ w B *Y 3 > B
p p t _ w < * 0 D ' = + 4 8 ?R C B B C B # p p t _ h B *Y 3 > B
p p t _ h < * 0 D ' = +K 4 8 ? C B B # p p t _ x + ( c o s ( - 2 * p i * ( 1 - $ ) ) * - # p p t _ x - s i n ( - 2 * p i * ( 1 - $ ) ) * ( 1 - # p p t _ y ) ) * ( 1 - $ ) C B ? B *Y 3 > B
p p t _ x < * 0 D ' = +K 4 8 ? C B B # p p t _ y + ( s i n ( - 2 * p i * ( 1 - $ ) ) * - # p p t _ x + c o s ( - 2 * p i * ( 1 - $ ) ) * ( 1 - # p p t _ y ) ) * ( 1 - $ ) C B ? B *Y 3 > B
p p t _ y < * 0 +
0 ^N0 ~ v | 0 (
|
|
H `
|
# l o ? "6 @ N N N ? N `N
G Based on census data from 1987 to 1995, a model for the hourly wage w of people employed in the production of computers and related goods in the United States is
w = 0.409t + 10.74, where t is the number of years since 1987. According to this model, in what year will the hourly wage for these workers be approximately $18.10? V H D ^
|
f ֜ o ? "6 @ N N N ? N @`F
Q To solve, graph the equation and find the value of t when the value of w = 18.10. B R 2 3 H
| 0 h ? 33 _ _ _ P P T 1 0 i . @ + D= ' =
@B + &?
0 ^N0 - - 8 < , (
H 8 `
>
# l o ? "6 @ N N N ? N `
d w = 0.409t + 10.74 6 @
#
[ , $ D 0 N 0
0 N 0
0 n
0 " ` > 0 n 2
0 " ` 0 0 n 2
0 " ` 0
0 k
A EXAMPLE 5
2
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6
0e 0e
B| C D E F8 A @ A o 8c 8c
? 1 d 0u 0@T y2 N P ' p < ' p A) B C D| E | | " " E A 9 - * # $ . ~ 5 y = s E q N j U a ` _ j Z s U | N v E r 9 i 5 c - R I F < < @ s
"
0e @ @ A B C D EE F G H I J K5% L M N O P Q R S T U W YZ[ \ ]^_ ` ab N E 5% N E 5% N F
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0 h ? 33 _ _ _ P P T 1 0 t . @ + V~ D ' & =
@B D ' =
@B A ? %, ( < + O %, (
< + D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * 9 % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * 9 Ds ' = % ( D ' = % ( D+ ' =4 @B B
B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * < % ( D ' = -g 6 B f a d e *< 3 < * < D ' = % ( D8 ' =A @B B
B B 0B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * ; % ( D ' = -g 6 B f a d e *< 3 < * ; D ' = % ( D@ ' = % ( D ' =4 @B B B B % ( ) ) ) D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * % ( D ' = + 4 8 ?R C B B C B # p p t _ w B *Y 3 > B
p p t _ w < * D ' = + 4 8 ?R C B B C B # p p t _ h B *Y 3 > B
p p t _ h < * D ' = +K 4 8 ? C B B # p p t _ x + ( c o s ( - 2 * p i * ( 1 - $ ) ) * - # p p t _ x - s i n ( - 2 * p i * ( 1 - $ ) ) * ( 1 - # p p t _ y ) ) * ( 1 - $ ) C B ? B *Y 3 > B
p p t _ x < * D ' = +K 4 8 ? C B B # p p t _ y + ( s i n ( - 2 * p i * ( 1 - $ ) ) * - # p p t _ x + c o s ( - 2 * p i * ( 1 - $ ) ) * ( 1 - # p p t _ y ) ) * ( 1 - $ ) C B ? B *Y 3 > B
p p t _ y < * +8 +0 + ; 0 + KG
0 ^N0 |3 t3 p 3 h 3 (
h
h
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kh
# l D3 o ? "6 @ N N N ? N `
Write and graph a function for each side of the equation 18.10 = 0.409t + 10.74 from Extra Example 4. What is the
t-coordinate of the point where the two graphs intersect? @ F , 9 F
h
H B
h
C Y
H H B
h
C Y
H H B
h
C Y
/ H / H B
h
C Y
H H B
h
C Y
$ H $ H B
h
C Y
F H F H B
h
C Y
H H B
h
C Y
H
H B
h
C Y
H
H B
h
C Y
H H B
h
C Y
Q H Q H B
h
C Y
H H B
h
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H H B
h
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h
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h H h H B
h
C Q s Q
H B
h
C s
H B
h
C S
s S
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C
s
H B
h
C U s U
H B
h
C
s
H B
h
C W s W
H B
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C s
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h
C O s O
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h
C s
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h
C M s M
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h
C s
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h
C s
H B
h
C L s L
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h
C s
H B
h
C J s J
H B
h
C Y
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h
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h
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h B
3 jJ P
p X
h
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D E F jJ X
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H
h
f ? o ? "6 @ N N N ? N P
5 t 2
h
f C o ? "6 @ N N N ? N `
5 w 2 (
h
f G o ? "6 @ N N N ? N
P
L 0 4 8 10 12 14 16 18 20 2 \
h
0e 0e
B| C D E F8 @ o 8c 8c
? 1 d 0u 0@T y2 N P ' p < ' p A) B C D| E | | " " E A 9 - * # $ . ~ 5 y = s E q N j U a ` _ j Z s U | N v E r 9 i 5 c - R I F < < @ s
"
0e @ @ A B C D E F G H I J K5% L M N O P Q R S T U W YZ[ \ ]^_ ` ab N 5% N 5% N
5%
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f K o ? "6 @ N N N ? N P
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18
16
14
12
10 l
h
f P o ? "6 @ N N N ? N 0 P , $
0
d Graph w = 18.10. 8 2
h
0e 0e
B C D E F o 8c 8c
? 1 d 0u 0@T y2 N P ' p < ' p A) B C D| E | | @ " 0e @ @ A B C D EE F G H I J K5% L M N O P Q R S T U W YZ[ \ ]^_ ` ab N E 5% N E 5% N F
5%
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0
h
f U o ? "6 @ N N N ? N P P
p , $
0
Graph w = 0.409t + 10.74 V 2
h
0e 0e
BL C7 D E F o 8c 8c
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5%
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0 2
h
# l o ? "6 @ N N N ? N Br , $ D 0
h
f t] o ? "6 @ N N N ? N
, $
0
| 4 The t coordinate of the point of intersection is 18. , 5 2 0 H
h 0 h ? 33 _ W _ _ _ P P T 1 0 7 . @ + Dc ' =
@B D ' =
@B A ? %, ( < + O %, (
< + D ' = % ( D ' = % ( DD ' =A @B B B B 0B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * h % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * h D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * h % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * h D ' = % ( D ' = % ( DD ' =A @B B B B 0B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * h % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * h Dr ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * h % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * h D ' = % ( D+ ' =4 @B B
B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * h % ( D ' = -g 6 B f a d e *< 3 < * h D ' = % ( D ' = % ( DD ' =A @B B B B 0B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * h % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * h + +0 + h 0 + +0 + h 0 + +0 + h 0 +
0 ^N0
m (
H x `
# l `y o ? "6 @ N N N ? N `~
h T h e a v e r a g e s p e e d s i n m i l e s p e r h o u r o f t h e f i r s t s i x w i n n e r s ( 1 9 1 1 1 9 1 6 ) o f t h e I n d i a n a p o l i s 5 0 0 c a n b e m o d e l e d b y t h e e q u a t i o n y = 1 . 8 4 x + 8 0 , w h e r e x i s t h e n u m b e r o f y e a r s s i n c e 1 9 1 1 a n d y i s t h e a v e r a g e s p e e d i n m i l e s p e r h o u r . U s e t h i s m o d e l t o f i n d t h e a v e r a g e s p e e d o f R a l p h D e P a l m a , t h e 1 9 1 5 w i n n e r . l 5
' u
f ( o ? "6 @ N N N ? N p , $ 0
] Regardless of the way you choose to solve this problem, the solution is 87.36 miles per hour. ^ ( 2 ^ H
0 h ? 33 z _ _ _ P P T 1 0 Z . @ + 2S D ' =
@B D ' =
@B A ? %, ( < + O %, (
< + D ' = % ( D ' = % ( D8 ' =A @B B
B B 0B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < *
% ( D ' = -g 6 B f a d e *< 3 < *
+8 +0 +
0 +
0 ^N0 p A (
N o ? "0 @ N N N ? N
C QUESTION:
( 2
q
f p o ? "6 @ N N N ? N @@
; What linear function is related to the equation
4x + 3 = 5? F 0 ( 2 1
N o ? "0 @ N N N ? N
A ANSWER: ( 2
f o ? "6 @ N N N ? N @@
, $ 0
y = 4 x 2 X H
0 3 h ? 33 _ _ _ P P T 1 0 . { + , DD ' =
@B D ' =
@B A ? %, ( < + O %, (
< + D6 ' = % ( D ' = % ( D ' =A @B B 5 B B 0B % ( ) ) ) D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * % ( D ' = + 4 8 ?R C B B C B # p p t _ w B *Y 3 > B
p p t _ w < * D ' = + 4 8 ?R C B B C B # p p t _ h B *Y 3 > B
p p t _ h < * D ' = -g 6 B f a d e *< 3 < * +8 +0 + 0 + 0 d (
d X
d C \
d S l \
0
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