2.3 Solving Equations

By Proof Justifications

Standards :

A .REI

.1

ARE 1.3

Old Solving Linear Equations

103×+4=31

202×+57×305×+5=-5×-53×+4=4: 31-4 2*2×+5-

7×-2×5×+55=5×-5-53x=27 5--5×5=-5×-10$ J 5¥ 5×+51=-5*5×10×=9 1=X

.

10×-10# to

×= -1.

nehdsolvingequatimby Proof Justification

Let's consider the following words : commute,associative

, reflectivesymmetric ,& transitive

.

• commute :

totravelbaoklforthCexampHThaddiusCommutes_fromLSHStohishomelhishometoLSHS.assoaahve-tjolnagroupCexamp1efTerinIMeKivahassoa_atewithJames.refleaive-mirrorimageslikereftedionKxamp1e1WhenAlissa1ooksinthemlrrir.sheseestheexaotsaMecomponentsaswhatsinreality.Symmetric-WhenoutinhalftheobjeothasthesamecomponentKxampH@h.transit-passingthnughKxamp1efJamiahusestheMartatgettothemakmovies.a

ndhome.

Let's consider a,b and c as Variables

.

Properties of OPERATIONS

I.

Commutative property of Addition at b = bta2

.

Commutative property of Multiplication a . b = b .a.

3.

Associative Property of Addition at ( btc ) = bt ( at )4. Associative Property of Multiplication a . ( b . c) = b • ( a . c )

.

5. Additive Inverse Property a +ta ) = 0

6. Multiplicative Inverse Property a • ±a = 1

.

7. Distributive Property albtc ) = abtac

Properties of EQUALITIES1

.Reflective Propert a = a

2. Symmetric Property If a=b

,then b=a

.

3.tt#hrePnpertylfa=b,b=c,then a =c

.

4.Addition Property of Equality If a=b

,then Atc = btc

5.Subtraction Property of Equality If a=b

,then at = b- c

.

6. Multiplication Property of Equality Hats ,

then a •c=b . c

7. Division Property of Equality If a :b,then

cd = §,

Examples] Use properties to solve the steps of solving equation .

����������� ������������������ !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~��������������������������������� ¡¢£¤¥¦§¨©ª«¬­®¯°±²³´µ¶·¸¹º»¼½¾¿ÀÁÂÃÄÅÆÇÈÉÊËÌÍÎÏÐÑÒÓÔÕÖ×ØÙÚÛÜÝÞßàáâãäåæçèéêëìíîïðñòóôõö÷øùúûüýþÿ1����������� ������������������ !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~��������������������������������� ¡¢£¤¥¦§¨©ª«¬­®¯°±²³´µ¶·¸¹º»¼½¾¿ÀÁÂÃÄÅÆÇÈÉÊËÌÍÎÏÐÑÒÓÔÕÖ×ØÙÚÛÜÝÞßàáâãäåæçèéêëìíîïðñòóôõö÷øùúûüýþÿ. 3×+4=31• given

.

3×+44=31-4• subtraction property

• }§=§• division property

• X = 9 • answer.20.2×+5=11

• given• 2*2×+5=7×-4 . subtraction property

•

§=5¥• division property

• 1=x . answer.

@•6(× -3=30 • given• 6×-12=30 • distributive property

•

6×-1*12=30+12

• addition property• ¥461 • division property

• ×=7.

• answer.

Exampled Namethecorreotpnperlyof operation .

D ( 51161=16115 ) commutative property of multiplication

D -1+1=0 additive inverse

D x=3and3=× symmetric property

@ 6+10=10+6 commutative properly of addition& 2+(4+5)=4+(2+5) associative property of additionµ 21×+51=2×+10 distribution property

D 5°¥=1 multiplicative inverse

Special cases for solving equations

CASEIE: one solution 11+5=10Xts -5=10-5

× =5.

CASEIN solution ×t6=× -5×t#6=× -5-6×=× -11

× - × =*x -11Ox = -11

OF-11

CHET infinite amount of solutions ×t6=xt6( reflective property )