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F.1 Mathematics Supplementary Notes Chapter 2 Using Algebra to Solve Problems 11/21 P. 1

Chapter 2 Using Algebra to Solve Problems Name!"""""""""""""# $ Class! F.1 # $

%mportant &erms

sum 和

term 項

'i((erence 差 constant term 常數項

pro'uct 積 li)e term 同類項

*uotient 商 unli)e term 異類項

algebraic e+pression 代數式 se*uence 序列

substitution 代入 general term 通項

symbol 符號 (unction 函數

e*uation 方程 variable 變數

ine*uality 不等式

number pattern 數型

,evision Notes

A. Using algebraic language

Symbol Meaning -+amples

＝ is e*ual to，e*uals －2＝

≠ is not e*ual to 0≠

is appro+imately#近似$ e*ual to

21 .

! is greater than，is larger than 3!

" is less than，is smaller than 4"1

because

there(ore

# x＝

x＝2

Exercise A

5evel 1

1. 6rite the (ollo7ing using symbols$

#a$ &he pro'uct o( 2 an' b e*uals 0 """""""""""""""""""""""""

#b$ m is less than 8 """""""""""""""""""""""""

#c$ a is not e*ual to 9ero """""""""""""""""""""""""

#'$ 2 is greater than :2 """""""""""""""""""""""""

#e$ 1.8:1.8 is appro+imately e*ual to 2 """""""""""""""""""""""""

#($ &he sum o( a an' b e*uals 1 """""""""""""""""""""""""

#g$ 6hen x is subtracte' (rom y，the

'i((erence is greater than 0 """""""""""""""""""""""""

#h$ &he *uotient 7hen d is 'ivi'e' by 12

is not e*ual to 9ero """""""""""""""""""""""""

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#i$ &he pro'uct o( p an' 2 is less than q """""""""""""""""""""""""

#;$ &7ice y e*uals 18，there(ore y e*uals to """""""""""""""""""""""""

"""""""""""""""""""""""""

#)$ x plus e*uals t7enty，there(ore x e*uals 1 """""""""""""""""""""""""

"""""""""""""""""""""""""

#l$ y minus 20 e*uals 8，there(ore y e*uals """""""""""""""""""""""""

"""""""""""""""""""""""""

5evel 2

2. 6rite the (ollo7ing using symbols$

#a$ &he pro'uct o( a an' b is greater then 2 to the po7er 8 """""""""""""""""""""""""

#b$ &he sum o( x an' e*uals q times p s*uare' """""""""""""""""""""""""

#c$ 6hen a is subtracte' (rom times o( b，the

'i((erence is less than hal( c """""""""""""""""""""""""

#'$ Four times the sum o( x an' y is greater than

the sum o( hal( a plus b """""""""""""""""""""""""

#e$ &he *uotient 7hen p is 'ivi'e' by one<thir'

#%&'$ q is not e*ual to r cube' """""""""""""""""""""""""

#($ c s*uare' plus 3 is less than the pro'uct o( d

an' t7o<thir's#%&($ d """""""""""""""""""""""""

=. Algebraic e+pressions

An algebraic e+pression can be 'ivi'e' into several parts by the >?> sign or the >@>sign.

-ach part together 7ith the >?> or >@>sign be(ore it is calle' a term o( the algebraic e+pression.

Substitution o( algebraic e+pressions

-+ample! Substitute x＝ the into the algebraic e+pression 12 2

+− x x 6hen x＝ the value o( the algebraic e+pression

12 2

+− x x ＝ 1$/#$/#2 2 +−

＝ 80

-+ample! 6hen x＝ @1， y＝ @2， z ＝，(in' the value o( yz x x

y x/2

+−−

Substitute x＝ @1， y＝ @2， z ＝ into yz x x

y x/2

+−−

then

yz x x

y x/2

+−−

$$#2#/$1#1

$2#$1# 2 −+−−−

−−−=

x is calle' the

variable.

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F.1 Mathematics Supplementary Notes Chapter 2 Using Algebra to Solve Problems 11/21 P.

B @1 @ 1 @

B @2

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C. Formulas

6e have learnt many mathematics (ormulas at primary level an' some o( them are sho7n belo7!

Figure Forumla%n 7or's Algebraic -+pression

S*uare#)方*$

length#+$area＝length o( one si'e : length o( one si'e

perimeter ＝ 8:length o( one si'e

5et the area be A

the perimeter be P

length o( one si'e be b

A＝ 2b

P ＝ 8b

,ectangle#+方*,-*$

length#+$

7i'th

#.$

area＝length : 7i'th

perimeter ＝2:#length#7i'th$

5et the area be A

the perimeter be P

the length be b

the 7i'th be w

A ＝ bw

P ＝ 2 #b# w$

&riangle#%/*$

height#0$

base#1$

area＝2

heightbase×

5et the area be A

the base be b

the height be h

A ＝ 2

bh

Circle#2$

ra'ius#34$ area＝ ra'ius : ra'ius : π

circum(erence#25$＝ 2 : ra'ius : π

5et the area be A

the ra'ius be r

the circum(erence be C

A ＝ π 2

r

C ＝ 2 πr

Exercise B5evel 1

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1. Complete the (ollo7ing table

Algebraic e+pression Number o( terms Constant term

#a$ 212 +÷+− x y x

#b$ ba −÷×+− /0

#c$n

mnm 3++

#'$ /44

12/+− kl

h

#e$ .2 +−+ y xy x

2. 6rite 'o7n a pair o( li)e terms in each o( the (ollo7ing algebraic e+pressions.

#a$ xy x y x 02

12 −−+ """""""""""""""""""""""""

#b$ xy y x yx −−+ 22 """""""""""""""""""""""""

#c$ aabba

++−− 20

"""""""""""""""""""""""""

#'$ y x x

+−−−

/

1

1/ """""""""""""""""""""""""

#e$ cbcabcababc −+−+− 2 """""""""""""""""""""""""

. 6hen x＝，(in' the values o( the (ollo7ing algebraic e+pressions.

#a$ 2 $# x x − #b$ /

2 2−+ x x

8. 6hen y＝，(in' the values o( the (ollo7ing algebraic e+pressions.

#a$ 2 y#－ y$2 #b$ y2## y－1$

. %n the (ormula 2/

3+= C F ， 2. %n the (ormula

2

.

22r A = ，

(in' the value o( F i( C ＝. (in' the value o( A i( r ＝

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0. iven the values a＝2，b＝，(in' the value o( x in the (ollo7ing

#a$ 2ab x = #b$ $#2

1ab x −=

#c$1−

=b

a x #'$ $2# += ba x

#e$ 2$#2

1ba x += #($ abb x −=

2

5evel 2

. 6hen x＝1， y＝ @2， z ＝，(in' the values o( the (ollo7ing algebraic e+pression

#a$ x－ y2# z #b$ # x2－ z $－2 x

#c$ # x－ y#2 z $2 #'$ #2 x－ y$# y# z $

#e$ #8 x2－ z $ y #($ 2 yx# xz －

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4. %(2

1= x ，

1= y ， 2= z ，(in' the values o( the (ollo7ing

#a$ y z y x −+ #b$ + y

z y x 1

3. iven that x＝ @2， y＝ an' z ＝，(in' the value o( A in each o( the (ollo7ing

#a$ x y z A 8/ ÷×= #b$ z y x A $# +=

#c$ 2

22

$# z y

y x A

+

+= #'$

z

x y A

2

22−

=

#e$ $2$## z y y x A +−= #($ $#2

y x x z y A

+− +=

5evel

#g$

z x

y

y x A

+

−=

11

#h$

x y

z x

z A

−

−=

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D. -*uations an' %ne*ualities

-*uations #a$ An e*uation contains an e*ual sign EB an' also one or more un)no7ns.

#b$ &o solve an e*uation is to (in' the value o( the un)no7n in that e*uation. &hat

value is calle' the solution o( the e*uation.

#c$ 6e usually solve e*uations using the metho' o( Eper(orming the same operation

on each si'e o( the e*uation.

%ne*ualities #a$ -+pressions that involve ine*uality signs E! an' E" are calle' ine*ualities.

#b$ All values that satis(y an ine*uality are calle' solutions o( that ine*uality.

#c$ Similar to solving e*uations 7e solve ine*ualities using the the metho' o(

Eper(orming the same operation on each si'e o( the ine*uality .

-. Number Patterns

A group o( numbers 7hich is arrange' in a certain pattern is calle' a sequence.

-ach number in the se*uence is calle' a term.

-+ample! 2 4 11 GG

(irst term ! 2 B #1$ @ 1 HB 1a B 2

secon' term ! B #2$ @ 1 HB 2a B

thir' term ! 4 B #$ @ 1 HB a B 4 ! !

! !

the n th term B #n$ @ 1 H B na B n @ 1

F. Functions

A (unction is li)e a number<machine. 6henever 7e input a number the machine 7ill output a

correspon'ing unmber.

Exercise C

1. -+press each the (ollo7ing sentence by an ine*uality an' test 7hether 14 is a solution by

the generalterm o( the

se*uence

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substitution.

Sentence %ne*uality Does 14 satis(y this

ine*ualityI

#a$ 6hen a is subtracte' (rom 1the 'i((erence is less than 2.

#b$ &he sum o( @3 an' 8 times o( x

is greater than 0.

#c$ 6hen 4 is subtracte' (rom2

1

o( x the result is greater than .

2. Solve the (ollo7ing ine*ualities.

#a$ x ?1! #b$ 2 x @ # x ? 2$"

"""""""""""""""""""""""""" """""""""""""""""""""""""""

"""""""""""""""""""""""""" """""""""""""""""""""""""""

"""""""""""""""""""""""""" """""""""""""""""""""""""""

"""""""""""""""""""""""""" """""""""""""""""""""""""""

#c$ 8 x ? #2@ x $!12 #'$ ? #+ @1$" 2 ? 8+

"""""""""""""""""""""""""" """""""""""""""""""""""""""

"""""""""""""""""""""""""" """""""""""""""""""""""""""

"""""""""""""""""""""""""" """""""""""""""""""""""""""

"""""""""""""""""""""""""" """""""""""""""""""""""""""

. 6rite 'o7n the ne+t three terms in the (ollo7ing se*uences

#a$ 2 4 11 GG """""""""""""""""""""""""

#b$ 1 @2 @8 GG """""""""""""""""""""""""

#c$ 1 @ GG """""""""""""""""""""""""

#'$ 2 8 4 10 GG """""""""""""""""""""""""

#e$2

1 0

1 3

1 12

1 GG """""""""""""""""""""""""

8. %n each o( the (ollo7ing se*uences (in' the (irst terms o( that se*uence by substituting 1 2 8

an' respectively (or n.

#a$ 1 −= nan #b$ nan 2/ −=

"""""""""""""""""""""""""" """""""""""""""""""""""""""

#c$n

an21= #'$

1+=

nnan

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"""""""""""""""""""""""""" """""""""""""""""""""""""""

#e$ nnan −= 2 #($ ( ) $1#1 +×−= na

n

n

"""""""""""""""""""""""""" """""""""""""""""""""""""""

JKptiontal

. -+press the nth term o( the (ollo7ing se*uences in terms o( n.

#a$ /8A8A2A21 ×××× """"""""""""""""""""""""""

#b$ 1A0AA1 """"""""""""""""""""""""""

#c$ 1A1A1A1 −− """"""""""""""""""""""""""

#'$ .A/AA1 −− """"""""""""""""""""""""""

#e$

1811

1A

114

1A

4/

1A

/2

1

⋅⋅⋅⋅ """"""""""""""""""""""""""

0. Consi'er the se*uence

8

/A

8A

2

A2

#a$ 6rite 'o7n the ne+t t7o terms o( the se*uence. """"""""""""""""""""

#b$ -+press the nth term o( the se*uence in terms o( n. """"""""""""""""""""

#c$ Use the result o( part #b$ to (in'

#i$ the th term """"""""""""""""""""

#ii$ the 1st term. """"""""""""""""""""

常678$http!//777.puiching.e'u.h)/Lmaths/stu'ent/

數9:;<=$

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>?$@數A&B

CDEF，GHIJKLM式6FNO@數，PQ$ 812

+− nn ，RHSTU n等V

，1，2，GGWM式IXY'Z@數，[WM式\]^A，_`aaKb'Z n A數c代入d

e不fYb'Z@數g

h;，108i數9jkl#Fermat$Em 12

2 +=n

n F nopAI\@數，qmrdesU n等

V ，1，2，，8IXYL'Z@數，nErthuJ，[差不v'wi>#12$xy數9jz

{STU n B |}不fNO'Z@數，qmrdeb 1/

/

221

22 +=+= F B xy ~•A x y &

€\'Z 數‚'Z 數，_fdeb x , y A數cƒI

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