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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

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 .

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 .

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 .

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 .

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

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 .

Practice set a

Solve 6 a = 42 for a .

a = 7

Solve 12 m = 16 for m .

m = 4 3

Solve y 8 = 2 for y .

y = 16

Solve 6.42 x = 1.09 for x .

x = 0.17 (rounded to two decimal places)

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

Solve 5 k 12 = 2 for k .

k = 24 5

Solve a b 2 c = 4 d for b .

b = - 8 c d a

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

y = 12 h

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

m = 15 p q k 2

Exercises

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

3 x = 42

x = 14

5 y = 75

6 x = 48

x = 8

8 x = 56

4 x = 56

x = 14

3 x = 93

5 a = 80

a = 16

9 m = 108

6 p = 108

p = 18

12 q = 180

4 a = 16

a = 4

20 x = 100

6 x = 42

x = 7

8 m = 40

3 k = 126

k = 42

9 y = 126

x 6 = 1

x = 6

a 5 = 6

k 7 = 6

k = 42

x 3 = 72

x 8 = 96

x = 768

y 3 = 4

m 7 = 8

m = 56

k 18 = 47

f 62 = 103

f = 6386

3.06 m = 12.546

5.012 k = 0.30072

k = 0.06

x 2.19 = 5

y 4.11 = 2.3

y = 9.453

4 y 7 = 2

3 m 10 = 1

m = 10 3

5 k 6 = 8

8 h 7 = 3

h = 21 8

16 z 21 = 4

Solve p q = 7 r for p .

p = 7 r q

Solve m 2 n = 2 s for n .

Solve 2.8 a b = 5.6 d for b .

b = 2 d a

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

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

b = 15 c 8

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

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

t = 3 p 2 4

Solve The product of a rectangle and a star over a triangle is equal to a rhombus. for .

Solve 3 Δ 2 = Δ for .

= 2 3

Exercises for review

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

( [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

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

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

all real numbers except 7

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

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
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
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
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
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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
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I'm interested in nanotube
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Source:  OpenStax, Algebra i for the community college. OpenStax CNX. Dec 19, 2014 Download for free at http://legacy.cnx.org/content/col11598/1.3
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