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Solve 3 ( m 6 ) 2 m = 4 + 1 for m .

3 ( m 6 ) 2 m = 4 + 1 3 m 18 2 m = 3 m 18 = 3 m = 15

C h e c k : 3 ( 15 6 ) 2 ( 15 ) = 4 + 1 Is this correct? 3 ( 9 ) 30 = 3 Is this correct? 27 30 = 3 Is this correct? 3 = 3 Yes, this is correct .

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Practice set b

Solve and check each equation.

16 x 3 15 x = 8 for x .

x = 11

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4 ( y 5 ) 3 y = 1 for y .

y = 19

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2 ( a 2 + 3 a 1 ) + 2 a 2 + 7 a = 0 for a .

a = 2

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5 m ( m 2 a 1 ) 5 m 2 + 2 a ( 5 m + 3 ) = 10 for a .

a = 10 + 5 m 6

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Often the variable we wish to solve for will appear on both sides of the equal sign. We can isolate the variable on either the left or right side of the equation by using the techniques of Sections [link] and [link] .

Sample set c

Solve 6 x 4 = 2 x + 8 for x .

6 x 4 = 2 x + 8 To isolate x on the left side, subtract 2 m from both sides . 6 x 4 2 x = 2 x + 8 2 x 4 x 4 = 8 Add 4 to both sides . 4 x 4 + 4 = 8 + 4 4 x = 12 Divide both sides by 4. 4 x 4 = 12 4 x = 3

C h e c k : 6 ( 3 ) 4 = 2 ( 3 ) + 8 Is this correct? 18 4 = 6 + 8 Is this correct? 14 = 14 Yes, this is correct .

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Solve 6 ( 1 3 x ) + 1 = 2 x [ 3 ( x 7 ) 20 ] for x .

On left side of an equation, arrows show that six is multiplied with each term inside the parentheses, and on right side, arrows show that three is multiplied with each term inside the parentheses.

6 18 x + 1 = 2 x [ 3 x 21 20 ] 18 x + 7 = 2 x [ 3 x 41 ] 18 x + 7 = 2 x 3 x + 41 18 x + 7 = x + 41 To isolate x on the right side, add 18 x to both sides . 18 x + 7 + 18 x = x + 41 + 18 x 7 = 17 x + 41 Subtract 41 from both sides . 7 41 = 17 x + 41 41 34 = 17 x Divide both sides by 17. 34 17 = 17 x 17 2 = x Since the equation 2 = x is equivalent to the equation x = 2 , we can write the answer as x = 2. x = 2

C h e c k : 6 ( 1 3 ( 2 ) ) + 1 = 2 ( 2 ) [ 3 ( 2 7 ) 20 ] Is this correct? 6 ( 1 + 6 ) + 1 = 4 [ 3 ( 9 ) 20 ] Is this correct? 6 ( 7 ) + 1 = 4 [ 27 20 ] Is this correct? 42 + 1 = 4 [ 47 ] Is this correct? 43 = 4 + 47 Is this correct? 43 = 43 Yes, this is correct .

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Practice set c

Solve 8 a + 5 = 3 a 5 for a .

a = 2

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Solve 9 y + 3 ( y + 6 ) = 15 y + 21 for y .

y = 1

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Solve 3 k + 2 [ 4 ( k 1 ) + 3 ] = 63 2 k for k .

k = 5

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Recognizing identities and contradictions

As we noted in Section [link] , some equations are identities and some are contradictions. As the problems of Sample Set D will suggest,

    Recognizing an identity

  1. If, when solving an equation, all the variables are eliminated and a true statement results, the equation is an identity.

    Recognizing a contradiction

  1. If, when solving an equation, all the variables are eliminated and a false statement results, the equation is a contradiction.

Sample set d

Solve 9 x + 3 ( 4 3 x ) = 12 for x .

On left side of an equation, arrows show that three is multiplied with each term inside the parentheses.

9 x + 12 9 x = 12 12 = 12

The variable has been eliminated and the result is a true statement. The original equation is an identity.

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Solve 2 ( 10 2 y ) 4 y + 1 = 18 for y .

On left side of an equation, arrows show that negative two is multiplied with each term inside the parentheses.

20 + 4 y 4 y + 1 = 18 19 = 18

The variable has been eliminated and the result is a false statement. The original equation is a contradiction.

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Practice set d

Classify each equation as an identity or a contradiction.

6 x + 3 ( 1 2 x ) = 3

identity, 3 = 3

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8 m + 4 ( 2 m 7 ) = 28

contradiction, 28 = 28

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3 ( 2 x 4 ) 2 ( 3 x + 1 ) + 14 = 0

identity, 0 = 0

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5 ( x + 6 ) + 8 = 3 [ 4 ( x + 2 ) ] 2 x

contradiction, 22 = 6

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Exercises

For the following problems, solve each conditional equation. If the equation is not conditional, identify it as an identity or a contradiction.

2 y + 7 = 3

y = 5

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5 x + 6 = 9

x = 3

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10 y 3 = 23

y = 2

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m 11 15 = 19

m = 44

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7 x 4 + 6 = 8

x = 8

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3 k 14 + 25 = 22

k = 14

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3 ( x 6 ) + 5 = 25

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16 ( y 1 ) + 11 = 85

y = 5

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23 y 19 = 22 y + 1

y = 20

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8 k + 7 = 2 k + 1

k = 1

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2 ( x 7 ) = 2 x + 5

contradiction

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4 ( 5 y + 3 ) + 5 ( 1 + 4 y ) = 0

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3 x + 7 = 3 ( x + 2 )

x = 3

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4 ( 4 y + 2 ) = 3 y + 2 [ 1 3 ( 1 2 y ) ]

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5 ( 3 x 8 ) + 11 = 2 2 x + 3 ( x 4 )

x = 19 14

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12 ( m 2 ) = 2 m + 3 m 2 m + 3 ( 5 3 m )

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4 k ( 4 3 k ) = 3 k 2 k ( 3 6 k ) + 1

k = 3

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3 [ 4 2 ( y + 2 ) ] = 2 y 4 [ 1 + 2 ( 1 + y ) ]

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5 [ 2 m ( 3 m 1 ) ] = 4 m 3 m + 2 ( 5 2 m ) + 1

m = 2

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For the following problems, solve the literal equations for the indicated variable. When directed, find the value of that variable for the given values of the other variables.

Solve I = E R for R . Find the value of R when I = 0.005 and E = 0.0035.

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Solve P = R C for R . Find the value of R when P = 27 and C = 85.

R = 112

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Solve z = x x ¯ s for x . Find the value of x when z = 1.96 , s = 2.5 , and x ¯ = 15.

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Solve F = S x 2 S y 2 for S x 2 S x 2 represents a single quantity. Find the value of S x 2 when F = 2.21 and S y 2 = 3.24.

S x 2 = F · S y 2 ; S x 2 = 7.1604

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Solve p = n R T V for R .

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Solve x = 4 y + 7 for y .

y = x 7 4

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Solve y = 10 x + 16 for x .

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Solve 2 x + 5 y = 12 for y .

y = 2 x + 12 5

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Solve 9 x + 3 y + 15 = 0 for y .

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Solve m = 2 n h 5 for n .

n = 5 m + h 2

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Solve t = Q + 6 P 8 for P .

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Solve A star = + 9 j Δ for j .

j is equal to the product of star and triangle minus square over nine.

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Exercises for review

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

( x + 3 ) 3 ( x 2 ) 7

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( [link] ) Find the product. ( x 7 ) ( x + 7 ) .

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( [link] ) Find the product. ( 2 x 1 ) 2 .

4 x 2 4 x + 1

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( [link] ) Solve the equation y 2 = 2.

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( [link] ) Solve the equation 4 x 5 = 3.

x = 15 4

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Questions & Answers

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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