2 Linear Law (Mas)
2 Linear Law (Mas)
2 Linear Law (Mas)
LINEAR LAW
Back to Basic 1 : How To Find Gradient of a straight line A linear equation can be written in the form y = ! " c . Note that is the gradient of the straight line, and c is the y!interce"t. #ou can find by using the gradient formula $ %
Example 1 y
A(&, .)
B(-, ')
O
%
x
/ 0
%
Your turn to try ! #1$ (ind the gradient of *+.
y
&#
1&
% %
*(', ,)
+(-, &) x
% % %
%
Linear law
'
MT (MAS)
Back to Basic % : How To Find y intercept and the equation of a straight line Ste" ' $ (ind gradient by using the gradient formula .
Ste" & $ 4rite the equation of straight line y = mx + c. Substitute your Ste" 5 $ (ind the y!interce"t c using any "oint on the line.
(ind the e()ation of the line A).
AB
Example 1
/ . ' % & 0
%*%
A(&, .)
mx + c 2x + c 2(2) + c 4 + c c
y 9 9 13
= = = =
B(-, ') x
Using A(2, 9 ,
y = 2x + 1! .
*(', ,)
+(-, &) x
6quation of *+ is 7777777777777..
R'
&
MT (MAS)
%ore &a'ic exerci'e' (or you ))** to &oo't your con(i+ence! +uestion $ (ind the equation of *+ in each of the following cases $ ' y & +(0, .)
. X
y . *(3, -)
5
+(5, 3) x
O
. X *(3,')
.
O
8 y = &x + ' 9
8 y = & x + - 9 0
5 y
. ;
+(-, '/)
.*(', -) ; .+(:,&) ;
. ;
*(!',0)
O
Linear law
MT (MAS)
8y % &; < -9
8y % 1 ; < ,9
,o#, let u' 'ee #-at come' out in S.% 2//)) LINEAR LAW
&a+er 1, #1-, 11, 1% -r 1. /0 mar1'0 " &a+er %, #1 /1/ mar1'0
&a+er 1 : 'hort #)estion o 2s)all3 a linear 4straight line5 gra+h is gi6en o Also gi6en is a non7linear e()ation o 8o) will 9e ask to calc)late the 6al)es of two )nknowns 9ased on the infor ation gi6en H:W to answer this ()estion : 4rite the linear equation in the form 8 = ; " < . Note that is the gradient of the straight line, and < is the 8!interce"t. #ou can find by using the gradient formula $
=f the line cuts the >ertical a;is, then c is easily obtained by reading the 8 ! interce"t. ?therwise use 8 = ; " < with the >alues of ; and 8. =: N:T use the non!linear equation for substitution of @ and #A
Example 1 S.% 2//! .aper 1, 21/ 3 x an$ y are related by the equation 3 = +!% " (!, where " and q are constants. A straight line is obtained by "lotting yB; against ; as shown in the diagram below. Calculate the >alues of " and q . mar%s9 84
Solution 3 Step 1& '(ange non linear e4uation y = )x & + qx to a linear equation
y = ) ( x) + q x
8 = Step 2& Step !& ; " <
p =
Linear law
MT (MAS)
1 = * 1% " ( 1. = (
4 = 1!
Now is 3o)r t)rn to tr3 so e ()estions of the sa e t3+e, ok ?
Exerci'e 1 S.% /! , 78O,E 21 3 x an$ y are related by the equation 3 = h!% " k!, where h and D are constants. A straight line is obtained by "lotting yB; against ; as shown in the diagram below. Calculate the >alues of h and D . 84 mar%s9 y1x
(', ,)
(-, &)
O
Solution 3 Step 1& '(ange non linear e4uation y = (x & + %x to a linear equation $
S.% /! , 78O,E 22 3 x an$ y are related by the equation 3 = (! " +!%, where " and q are constants. A straight line is obtained by "lotting yB; against ; as shown in the diagram below. Calculate the >alues of " and q . 84 mar%s9 y1x Answer $
(/, -)
(3, &)
O
Linear law
MT (MAS)
Now, Eet us "ractise drawing 9t-e line o( &e't (it:3 '.' Fraw the line of best fit ' y &
log '3 ;
;o# a&out (in+in5 t-e e4uation repre'entin5 t-e line o( &e't (it <
'.& (ind the equation for the line of best fit of the following gra"hs. ' y
X X X
+(:,'5)
&
. X
y . *(3,0)
X X X X X
. X*(3,5)
. ; Gradient, m %
+(&,3)
Linear law
MT (MAS)
8 y = &x + 5 9
8 y = & x + 0 9
*(!',0) ;&
. ;
+(:,'-) . ;
.*(',/) ; ; ; ; .+(5,&) ;
;&
8y % &;&<-9
8y% !5;&<''9
The diagram below shows the line of best fit for the gra"h of y& against ;. Fetermine the non!linear equation connecting y and ;.
y& X*(3,0)
y&
+(&,3) ;
+(&,'&) *(3,&)
03 3&
' x
; " <,
8 = 3 , ; = !,
Linear law
MT (MAS)
8y&%!&;<09 5 The diagram below shows the line of best fit for the gra"h of 0
& 8 y = : + & 9 x
'
y x&
X
X+(-,'&)
y x&
(&,0) 3
(0, :)
*(5,3)
; ;
8 :
y x
&
= 0 x '& 9 -
y x
&
The diagram below shows the line of best fit for the gra"h of xy against ;. Fetermine the relation between y and ;. ;y
X +(0,'&)
The diagram below shows the straight line gra"h of xy against ;. 6;"ress y in terms of ;. ;y (0,'3)
*(&,0) ;
(', ') ;
Linear law
MT (MAS)
8 y = 0
0 9 x
8 y = 5 9
& x
Vertical Axis
ost +oints5
F$ G2se 3o)r gra+h to find the 6al)es of @@H 'o, =: N:T calc)late )sing the original 6al)es fro the ta9leB
11$ 2se 3o)r gra+h to o9tain f)rther infor ation : to find the )nknowns @@@$
Linear law
MT (MAS)
(a) *lot !3 against !. using a 'cale o( 2 cm to 1 unit on t-e x!@axi' and 2 cm to 1/ unit' on t-e xy@axi'* Ience, draw the line of best fit *2mar%s . (b) (rom the gra"h, estimate the >alue of (i p an+ 4 0: (ii x #-en y = *2mar%s. x
8"ns & )= 01345, q = 92, x=14449
AN'WER F:R E;AL&LE 1 /<o+3 this ta9le onto the to+ +art of 3o)r gra+h +a+er0
!. !3
-$E F1$-
1$1I$-
%$% EI$1
%$1 E1$1
.$C .F$>
C$I 1.$I
(ollow ti"s ' to ''. Hsing the gra"h "a"er "ro>ided, "lot the "oints carefully and draw the line of best fit.
Re e 9er $ Hse a 'cale o( 2 cm to 1 unit on t-e x!@axi' and 2 cm to 1/ unit' on t-e xy@axi'* And , 3o)r scale L2'T BE 2NIF:RL B
q x ! xy = px + 4 y =
#
)x & +
m@
< C
m % " , and C % q
(rom your gra"h, find the gradient and read the >ertical interce"t. #ou should find that (i) ) 6 013435, q 6 92,
Linear law
'3
MT (MAS)
*lot the gra"h y a5ain't x2, using a 'cale o( 2 cm to 1 unit on t-e x2@axi' and 2 cm to /*B unit' on t-e y@axi'* 84 mar%s9 Hse the gra"h to e'timate the >alues of (i) " (ii) D y ) (iii) ; which satisfy the simultaneous equation = + %x and y % &. 83 mar%s9 x x
/An'#er3 p=B*1, 1 = @/*/0, x= A*6 A*A?
4a5 &lot log1- 3 against ! 93 )sing a scale of 2 cm to 2 unit' on the !7a!is and 2 cm to /*2 unit on the log1- 37a!is$ Hence, draw the line of 9est fit$ / 0 mar1' 0 (b) Hse your gra"h from (a) to find the >alue of (i) ) (ii) % 8 3 mar%s9
"nswer &) =14829, % =14399
495 &lot !3 against x2 , 93 )sing a scale of 2 cm to B )nits on 9oth a!es$ Hence draw the line of 9est fit$
Linear law
''
MT (MAS)
(c) Hse the gra"h from (a) to find the >alue of (i) ) (ii) r
4a5 &lot lo5 y against 4!"15, )sing a scale of % c on the log 37a!is$ Hence, draw the line of 9est fit$ (b) Hse you gra"h from (a) to find the >alues of (i) ) (ii) %
/ B mar1'0
82 mar%s9
Linear law
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