<< Chapter < Page Chapter >> Page >
This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. In this chapter, the emphasis is on the mechanics of equation solving, which clearly explains how to isolate a variable. The goal is to help the student feel more comfortable with solving applied problems. Ample opportunity is provided for the student to practice translating words to symbols, which is an important part of the "Five-Step Method" of solving applied problems (discussed in modules (<link document="m21980"/>) and (<link document="m21979"/>)). Objectives of this module: understand the equality property of addition and multiplication, be able to solve equations of the form ax = b and x/a = b.

Overview

  • Equality Property of Division and Multiplication
  • Solving a x = b and x a = b for x

Equality property of division and multiplication

Recalling that the equal sign of an equation indicates that the number represented by the expression on the left side is the same as the number represented by the expression on the right side suggests the equality property of division and multiplication, which states:

  1. We can obtain an equivalent equation by dividing both sides of the equation by the same nonzero number, that is, if c 0 , then a = b is equivalent to a c = b c .
  2. We can obtain an equivalent equation by multiplying both sides of the equation by the same nonzero number, that is, if c 0 , then a = b is equivalent to a c = b c .

We can use these results to isolate x , thus solving the equation for x .

Solving a x = b for x

a x = b a is associated with x by multiplication . Undo the association by dividing both sides by a . a x a = b a a x a = b a 1 x = b a a a = 1 and 1 is the multiplicative identity . 1 x = x

Got questions? Get instant answers now!

Solving x a = b for x

x = b a This equation is equivalent to the first and is solved by x . x a = b a is associated with x by division . Undo the association by multiplying both sides by a . a x a = a b a x a = a b 1 x = a b a a = 1 and 1 is the multiplicative identity . 1 x = x x = a b This equation is equivalent to the first and is solved for x .

Got questions? Get instant answers now!

Solving a x = b And x a = b For x

Method for Solving a x = b and x a = b

To solve a x = b for x , divide both sides of the equation by a .
To solve x a = b for x , multiply both sides of the equation by a .

Got questions? Get instant answers now!

Sample set a

Solve 5 x = 35 for x .

5 x = 35 5 is associated with x by multiplication . Undo the association by dividing both sides by 5 . 5 x 5 = 35 5 5 x 5 = 7 1 x = 7 5 5 = 1 and 1 is multiplicative identity . 1 x = x . x = 7

C h e c k : 5 ( 7 ) = 35 Is this correct? 35 = 35 Yes, this is correct .

Got questions? Get instant answers now!

Solve x 4 = 5 for x .

x 4 = 5 4 is asssociated with x by division . Undo the association by multiplying b o t h sides by 4. 4 x 4 = 4 5 4 x 4 = 4 5 1 x = 20 4 4 = 1 and 1 is the multiplicative identity . 1 x = x . x = 20

C h e c k : 20 4 = 5 Is this correct? 5 = 5 Yes, this is correct .

Got questions? Get instant answers now!

Solve 2 y 9 = 3 for y .

Method (1) (Use of cancelling):

2 y 9 = 3 9 is associated with y by division . Undo the association by multiplying b o t h sides by 9. ( 9 ) ( 2 y 9 ) = ( 9 ) ( 3 ) 2 y = 27 2 is associated with y by multiplication . Undo the association by dividing b o t h sides by 2. 2 y 2 = 27 2 y = 27 2

C h e c k : 2 ( 27 2 ) 9 = 3 Is this correct? 27 9 = 3 Is this correct? 3 = 3 Yes, this is correct .

Method (2) (Use of reciprocals):

2 y 9 = 3 Since 2 y 9 = 2 9 y , 2 9 is associated with y by multiplication . Then, Since 9 2 2 9 = 1 , the multiplicative identity, we can ( 9 2 ) ( 2 y 9 ) = ( 9 2 ) ( 3 ) undo the associative by multiplying b o t h sides by 9 2 . ( 9 2 2 9 ) y = 27 2 1 y = 27 2 y = 27 2

Got questions? Get instant answers now!

Solve the literal equation 4 a x m = 3 b for x .

4 a x m = 3 b m is associated with x by division . Undo the association by multiplying b o t h sides by m . m ( 4 a x m ) = m 3 b 4 a x = 3 b m 4 a is associated with x by multiplication . Undo the association by multiplying b o t h sides by 4 a . 4 a x 4 a = 3 b m 4 a x = 3 b m 4 a

C h e c k : 4 a ( 3 b m 4 a ) m = 3 b Is this correct? 4 a ( 3 b m 4 a ) m = 3 b Is this correct? 3 b m m = 3 b Is this correct? 3 b = 3 b Yes, this is correct .

Got questions? Get instant answers now!

Practice set a

Solve 6 a = 42 for a .

a = 7

Got questions? Get instant answers now!

Solve 12 m = 16 for m .

m = 4 3

Got questions? Get instant answers now!

Solve y 8 = 2 for y .

y = 16

Got questions? Get instant answers now!

Solve 6.42 x = 1.09 for x .

x = 0.17 (rounded to two decimal places)

Got questions? Get instant answers now!

Use a calculator to solve this equation. Round the result to two decimal places.

Solve 5 k 12 = 2 for k .

k = 24 5

Got questions? Get instant answers now!

Solve a b 2 c = 4 d for b .

b = - 8 c d a

Got questions? Get instant answers now!

Solve 3 x y 4 = 9 x h for y .

y = 12 h

Got questions? Get instant answers now!

Solve 2 k 2 m n 5 p q = 6 n for m .

m = 15 p q k 2

Got questions? Get instant answers now!

Exercises

In the following problems, solve each of the conditional equations.

5 a = 80

a = 16

Got questions? Get instant answers now!

6 p = 108

p = 18

Got questions? Get instant answers now!

4 a = 16

a = 4

Got questions? Get instant answers now!

6 x = 42

x = 7

Got questions? Get instant answers now!

8 m = 40

Got questions? Get instant answers now!

3 k = 126

k = 42

Got questions? Get instant answers now!

x 8 = 96

x = 768

Got questions? Get instant answers now!

m 7 = 8

m = 56

Got questions? Get instant answers now!

f 62 = 103

f = 6386

Got questions? Get instant answers now!

5.012 k = 0.30072

k = 0.06

Got questions? Get instant answers now!

y 4.11 = 2.3

y = 9.453

Got questions? Get instant answers now!

3 m 10 = 1

m = 10 3

Got questions? Get instant answers now!

8 h 7 = 3

h = 21 8

Got questions? Get instant answers now!

16 z 21 = 4

Got questions? Get instant answers now!

Solve p q = 7 r for p .

p = 7 r q

Got questions? Get instant answers now!

Solve m 2 n = 2 s for n .

Got questions? Get instant answers now!

Solve 2.8 a b = 5.6 d for b .

b = 2 d a

Got questions? Get instant answers now!

Solve m n p 2 k = 4 k for p .

Got questions? Get instant answers now!

Solve 8 a 2 b 3 c = 5 a 2 for b .

b = 15 c 8

Got questions? Get instant answers now!

Solve 3 p c b 2 m = 2 b for p c .

Got questions? Get instant answers now!

Solve 8 r s t 3 p = 2 p r s for t .

t = 3 p 2 4

Got questions? Get instant answers now!

Solve 3 Δ 2 = Δ for .

= 2 3

Got questions? Get instant answers now!

Exercises for review

( [link] ) Simplify ( 2 x 0 y 0 z 3 z 2 ) 5 .

Got questions? Get instant answers now!

( [link] ) Classify 10 x 3 7 x as a monomial, binomial, or trinomial. State its degree and write the numerical coefficient of each item.

binomial; 3rd degree; 10 , 7

Got questions? Get instant answers now!

( [link] ) Simplify 3 a 2 2 a + 4 a ( a + 2 ) .

Got questions? Get instant answers now!

( [link] ) Specify the domain of the equation y = 3 7 + x .

all real numbers except 7

Got questions? Get instant answers now!

( [link] ) Solve the conditional equation x + 6 = 2 .

Got questions? Get instant answers now!

Questions & Answers

how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Please keep in mind that it's not allowed to promote any social groups (whatsapp, facebook, etc...), exchange phone numbers, email addresses or ask for personal information on QuizOver's platform.
QuizOver Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask