Chapter 4 - Linear Equations in One Unknown

8/12/2019 Chapter 4 - Linear Equations in One Unknown

1/26

4 Linear Equations in One Unknown


Level 1 Questions

(P01C04L03Q041)

Solve the equation 23 =a an! "he"k the solution#

Solution

33$

33

$3

2323

=

=

=+=+

=

a

a

a

aa

Che"kin%& Su'stitute 3=a into 'oth si!es o the equation#L# #S# 2)3(3 ==*# #S# 2=

L# #S# = *# #S# 3=a is the solution#

(P01C04L03Q042)

Solve the equation 02+ = B an! "he"k the solution#

Solution

12+2

++

2+

2022+

02+

=

==

+=+=

B

B

B

B

B

Che"kin%& Su'stitute 12= B into 'oth si!es o the equation#

L# #S# 02)12(+ ==*# #S# 0=

L# #S# = *# #S# 12= B is the solution#

(P01C04L03Q043)

1


2/26

Question ,ank U-!ate 1.

Solve the equation 14/

=+ y an! "he"k the solution#

Solution

1/

/)3(//

3/

4144/

14/

=

=

=

=+

=+

y

y

y

y

y

Che"kin%& Su'stitute 1/= y into 'oth si!es o the equation#

L# #S# 14/1/ =+=

*# #S# 1= L# #S# = *# #S# 1/= y is the solution#

(P01C04L03Q044)

Solve the equation2

104

x= an! "he"k the solution#

Solution

41

2

2101010

10

22

1024

210

4

==

= =

==

=

x

x

x x

x

x

x

Che"kin%& Su'stitute41= x into 'oth si!es o the equation#

L# #S# 4=

2


3/26


*# #S#4

241

10=

=

L# #S# = *# #S#

41= x is the solution#

(P01C04L03Q04/)

Solve the equation 0121

2=+t an! "he"k the solution#

Solution

+1

2121

22

121

2

0121

2

=

=

=

=+

t

t

t

t

Che"kin%& Su'stitute+

1=t into 'oth si!es o the equation#

L# #S# 0121

2+1

=+

=

*# #S# 0= L# #S# = *# #S#

+1=t is the solution#

3


4/26


(P01C04L03Q04+)

Solve the equation 22#03 =+m an! "he"k the solution#

Solution

+#03#1

33

#13

2#022#02#03

22#03

=

=

==+

=+

m

m

m

m

m

Che"kin%& Su'stitute +#0=m into 'oth si!es o the equation#L# #S# 22#0)+#0(3 =+=*# #S# 2=

L# #S# = *# #S# +#0=m is the solution#

(P01C04L03Q04 )

Solve the equation x x 3$113 =+ #

Solution

211+

1+

1$1+

$11+

$1313

3$113

=

=

=

=

=+

=++

=+

x

x

x

x

x

x x

x x

(P01C04L03Q04 )

Solve the equation x x 2)1(3/ = #

4


5/26


Solution

$

31

3

33

213/

213

/

2)1(3/

=

=

=

+=+=+

=

x

x

x

x x

x x

x x

(P01C04L03Q04$)

Solve the equation 310

= cc #

Solution

310$

30

30$

3010

10310

10

310

=

=

==

=

=

c

c

c

cc

cc

cc

(P01C04L03Q0/0)

Solve the equation41+3

1 uu =+ #

/


6/26


Solution

441+

1+4

1+3

431

441+

3

1

41+

41+31

=

=

==+

=

+

=+

=+

u

u

u

uu

uu

uu

uu

(P01C04L03Q0/1)

Solve the equation 112)(3 =+ x x #

Solution

2

/10

10/211123

112213112)(3

=

=

=+=

==+

x

x

x

x x

x x

x x

(P01C04L03Q0/2)

Solve the equation 22)12(4)3(/ += y y #

Solution

133

33

1/1/

11//

2241//

22)12(4)3(/

==

=+= =

+=+=

y

y

y

y y

y y

y y

y y

(P01C04L03Q0/3)

Solve the equation

=4

331/

+1

p p#

+


7/26


Solution

1211

132

13211

132$2

1211124

3+

1124

3+

43

31/+1

43

31/+

1

==

==

=

=+=

=

p

p

p p

p p

p p

p p

p p

(P01C04L03Q0/4)

he su o x an! 1+ is 2/# in! the value o x#

Solution

.""or!in% to the -ro'le we "an set u- an al%e'rai" equation as ollows&

$

1+2/

2/1+

=

==+

x

x

x

(P01C04L03Q0//)

here are y a--les in a 'o5# he wei%ht o the 'o5 o a--les is 2 00 % an! the wei%ht o the e -t6

'o5 is 0 %# 7 ea"h a--le wei%hs 1+0 % in! the value o y#


8/26


Solution


121+0

1$20

1$201+0

02 001+0

2 0001+0

=

==

==+

y

y

y

y

y

(P01C04L03Q0/+)

here are lowers o three !i erent "olours in the %ar!en 2 n 8 1 o the are white n 9 2 are re!

an! 3 n are 6ellow# 7 there are 2/ lowers in the %ar!en in! the value o n#

Solution


4+

24

12/+

2/1+

2/3)2()12(

=

=

==+=+++

n

n

n

n

nnn

(P01C04L03Q0/ )here are $ rats in a 'arn# .t ni%ht our "ats A B C an! D enter the 'arn# Ea"h o "ats A B an! C

"at"hes y rats an! "at D "at"hes ( y 9 1) rats# .ll the $ rats are "au%ht# in! the value o y#

Solution


24

4

$14

$)1(3

=

=

==+=++

y

y

y

y

y y


9/26


(P01C04L03Q0/ )

here are x ale ra''its an! x e ale ra''its in a ar # Ea"h e ale ra''it or s a a il6 with

one ale ra''it# Ea"h a il6 %ives 'irth to a ra''it in a 6ear# 7 there are 4 ra''its in the ar the

ne5t 6ear what is the value o x:

Solution


234

43

42

=

=

==+

x

x

x

x x

(P01C04L03Q0/$)

hree onke6s A B an! C in! x 'ananas on a 'anana tree# ;onke6 A is the stron%est an! it takes

hal o the 'ananas while onke6s B an! C share the 'ananas le t equall6 an! one unri-e 'anana

is le t on the tree# 7 onke6 B has 2 'ananas in! the value o x#

Solution


10/26


(P01C04L03Q0+0)

7n 14$ a Portu%uese navi%ator a!e a vo6a%e ro Lis'on to the Ca-e o =oo! o-e in)224( + x !a6s at an avera%e s-ee! o / k >h# 7 the !istan"e "overe! was )3+03000( x k

how lon% !i! the vo6a%e take:

Solution


/120+00

+001202403+030002 0

3+030002402 03+03000)224(1203+03000)224(24/

==

===+=+=+

x

x

x

x x

x x

x x

x x

i e taken '6 the vo6a%e

!a6s122!a6s)2/24(

=

+=

10


11/26


Level 2 Questions

(P01C04L04Q031)

Solve the equation )1(2/#02/#0 x x = an! "he"k the solution#

Solution

3/#0

2/#2

2/#2/#0

22/#02/#0/#0

2/#02/#02/#0

)1(2/#02/#0

=

=

=+=+=

=

x

x

x

x x

x x

x x

Che"kin%& Su'stitute 3= x into 'oth si!es o the equation#L# #S# /#02)3(/#0 ==*# #S# /#0)31(2/#0 ==

L# #S# = *# #S# 3= x is the solution#

(P01C04L04Q032)

Solve the equation2

243 =+ t t an! "he"k the solution#

Solution

1212

443443

)2(4432

2432

243

= =

==+=+

=+

=+

t t

t t

t t

t t

t t

t t

Che"kin%& Su'stitute 12=t into 'oth si!es o the equation#L# #S# /

4)12(3 =+=

*# #S# /2

212 ==

L# #S# = *# #S# 12=t is the solution#

(P01C04L04Q033)

11


12/26


Solve the equation221

41

)34(20 a

a += an! "he"k the solution#

Solution

43

324243

243324

24034320

43240320

4214)+00(4

)21(21+00

221

41)34(20

=

=

=+=+=

+=

+=

+=

a

a

a

aa

aa

aa

aa

aa

Che"kin%& Su'stitute43=a into 'oth si!es o the equation#

L# #S# 0343

420 =

=

*# #S# 02

43

21

41 =

+=

L# #S# = *# #S#

43=a is the solution#

(P01C04L04Q034)

Solve the equation3

)12(24+ += x x #

Solution

11+1+1+1+

1241224121

)12(23)4+(3

)12(24+

==

==

+=+=

+=

x

x

x

x x

x x x x

x x

12


13/26


(P01C04L04Q03/)

wo trian%les have the sa e 'ase o y " one has a hei%ht o + " an! the other has a hei%ht o

" # 7 the areas o the two trian%les are !i ere! '6 2"10 in! the value o y#

Solution


101034

10)+)((21

))((21

===

y

y y

y y

(P01C04L04Q03+)

7n 1 4+ an ar"haeolo%ist oun! a u 6 in an an"ient to ' an! the a%e o the u 6 was)$$00$ +0( x 6ears# 7n 200+ the a%e o u 6 is two thousan! 6ears# ?hat is the value o x:

Solution


#0$$00

$20

$20$$00$$202000$$00

2000$$00$$20

20001+0$$00$ +0

2000)1 4+200+($$00$ +0

==

= =

==+=+

x

x

x x

x

x

x

(P01C04L04Q03 )

@urin% a warti e -erio! so e sol!ers were "a-ture! an! taken -risoners '6 their ene 6# 7n the

irst week hal o the sol!ers es"a-e!# al o the re ainin% sol!ers also es"a-e! in the se"on!

week# he last sol!ers es"a-e! in the thir! week an! no -risoner was le t# ?hat was the ori%inalnu 'er o sol!ers who were a!e -risoners:

13


14/26


Solution

Let N 'e the ori%inal nu 'er o sol!ers who were a!e -risoners#

32

)4)((4

442

42

221

2

221

2

==

=

=+

=+

=

+

=+

+

N

N

N

N N N

N N N

N N N

N N

N N

32 sol!ers were a!e -risoners#

(P01C04L04Q03 )

or the -ast 12 000 6ears the hei%ht o a ountain was in"rease! '6 0#4 " ea"h 6ear# . ter a

lan!sli!e the hei%ht o the ountain !ro--e! to 12 whi"h was32

o its hei%ht 'e ore the

lan!sli!e# ?hat was the hei%ht o the ountain 12 000 6ears a%o:

SolutionLet h 'e the hei%ht o the ountain 12 000 6ears a%o#

1020

410+

10+423

124

1232

)004#000012(

=

=

=+

=+

=+

h

h

h

h

h

he hei%ht o the ountain 12 000 6ears a%o was 1020 #

14


15/26


Level 2+ Questions

(P01C04L0/Q00 )

N is a 3A!i%it nu 'er# 7t is %iven that a N += $20 where a is an inte%er 'etween 0 an! $#

(a) 7 the or!er o the irst an! the thir! !i%its o the nu 'er is reverse! in! its new value#

(b) in! the !i eren"e 'etween the two 3A!i%it nu 'ers#

(c) in! the value o a i the !i eren"e is $$#

Solution

(a)


16/26


Solution

(a) (i)

31

/2

+/+/

2/

+2

2/

22/

+

22

3232

21

=

=

=

=

=+

=+++

x

x

x

x

x

x

(ii)

4

1

312

123

121

23

2312

123

4343

32

=

=

=

=

=+=

+++

x

x

x

x

x

x

(b) r6 1002= x #Su'stitute 1002= x into the le t han! si!e o the equation#

210021001

200310021001

20032

100210012003

10021001100110022

100210012003

10021

10011

2

100210012003

10021

11001

11

100210012003

10021001

10011000

L# #S#

=+=++=

+

+=

+

+

=

++=

2*# #S# = *# #S#L# #S# = 1002= x is the solution o the equation#

1+


17/26


18/26


(P01C04L0+Q033)

Solve the equation 4#22#0 =+ y #!" 2#1= y

#" 3= y

C" #4= y$" += y

Solution

@

+2#02#1

2#12#0

#242#0

4#22#0

=

=

=

=

=+

y

y

y

y

y

(P01C04L0+Q034)

Solve 33)+(13 =+ x x #!" /= x#" += xC" /= x$" += x

Solution

.

/210

102

33

33

33+13

33)+(13

=

=

==+

===+

x

x

x

x x

x x

x x

x x

1


19/26


(P01C04L0+Q03/)

Solve the equation3

)1(2/ t

t = #

!" 2=t #" 3=t C" 4=t $" /=t

Solution

,

3

212121+

+21++1/

)1(+1/33)B1(2/C3

3)1(2/

==

==

==+=

=

=

t

t

t

t t

t t

t t

t t

t t

t t

(P01C04L0+Q03+)

Solve the equation aa =+ +312 #

!" 3=a#" /=aC" 3=a$" /=a

1$


20/26


Solution

,

/$4/

4/$

34$

43$

)(+3$

+3$

+3

12+3

+3

12

=

=

=

+=

=

=

=

=+

=+

=

+

a

a

a

a

a

a

a

aa

aa

aa

(P01C04L0+Q03 )

Solve the equation r r

2#0$1#0

+= #

!" 10=r #" 12=r C" 14=r $" 1+=r

Solution

.

10#$

$

$#$

$2#010

2#0$10

2#0$1#0

=

===

+=

+=

r

r

r

r r

r r

r r

20


21/26


(P01C04L0+Q03 )

Solve the equation 214#03#02#0

+=+ nnn #

!" #1

=n

#" +#3=nC" #1=n$" +#3=n

Solution

@

+#3/

10

1221

21120

2112

3040+0

214

103

102

10

214

103

102

10

214#03#02#0

==

=

=

=+

=+

+=+

+=+

n

n

n

n

nnn

nnn

nnn

nnn

(P01C04L0+Q03$)

7 10 inus3

1 o x equals x inus 2 what is the value o x:

!" 3

#" +

C"$" $

21


22/26


Solution

@


$43

12

1234

10231

23110

=

=

=

=

=

x

x

x

x x

x x

(P01C04L0+Q040)

7 hal o c equals 2 inus3

1 o c what is the value o c:

!"12/

#"/

12

C" +

$" 12

Solution

,


/12

/+

2

2+/

2+

23

232

32

2

=

=

=

=+

=+

=

c

c

c

cc

cc

cc

(P01C04L0+Q041)

elen a!o-te! a 1A6earAol! !o% when she was x 6ears ol!# . ter $ 6ears her a%e was twi"e the a%e

22


23/26


o the !o%# in! the value o x#

!" $

#" 10

C"11

$" 12

Solution

C


11

$20

20$)$1(2$

=

==+

+=+

x

x

x

x

(P01C04L0+Q042)

7n a 'irth!a6 -art643

o a "ake was eaten# 1/0 % o the "ake was eaten in the se"on! !a6# Susan

ate the re ainin% "ake whi"h was0

1o the "ake in the thir! !a6# in! the ori%inal wei%ht o the

"ake#!" 1#2 k%

#" 1#4 k%C" 1#/ k%

$" 1# k%

Solution

.

Let the ori%inal wei%ht o the "ake 'e x %#

1200

)(1/0

1/01

1/0

1/0

11/0

43

=

=

=

=

=+

=++

x

x

x

x x

x x

x x x

he ori%inal wei%ht o the "ake is 1#2 k%#(P01C04L0+Q043)

23


24/26


Sue le t on% Don% or a holi!a6 tri-# She sta6e! 3 !a6s in


25/26


Solution

,


1/2

30

302

3//3

/33/

==

==

+=+

N

N

N

N N

N N

Lolita is 1/ 6ears ol!#

(P01C04L0+Q04/)

. train that travels ro ,eiGin% to ;os"ow sto-s at several "ities su"h as Ulaan'aatar 7rkutsk an!


26/26


Solution

@


132

12120

12012

43012

30)12(41

24/4)12(41

/424)12(41

=+=

==

=

=

=+

t

t

t

t

t

t

t

132 hours H / !a6s 12 hours he train arrives in ;os"ow in / !a6s 12 hours#

Documents

Chapter 4 - Linear Equations in One Unknown