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x + 1 = 10

conditional, x = 9

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y 4 = 7

conditional, y = 11

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5 a = 25

conditional, a = 5

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x 4 = 9

conditional, x = 36

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18 b = 6

conditional, b = 3

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y 2 = y 2

identity

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x + 4 = x 3

contradiction

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x + x + x = 3 x

identity

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8 x = 0

conditional, x = 0

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m 7 = 5

conditional, m = 2

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Literal equations

Literal equations

Some equations involve more than one variable. Such equations are called literal equations .

An equation is solved for a particular variable if that variable alone equals an expression that does not contain that particular variable.

    The following equations are examples of literal equations.

  1. y = 2 x + 7 . It is solved for y .
  2. d = r t . It is solved for d .
  3. I = p r t . It is solved for I .
  4. z = x u s . It is solved for z .
  5. y + 1 = x + 4 . This equation is not solved for any particular variable since no variable is isolated.

Solving equation of the form x + a = b and x a = b

Recall 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.

This is the this number same as number x = 6 x + 2 = 8 x 1 = 5

    This suggests the following procedures:

  1. We can obtain an equivalent equation (an equation having the same solutions as the original equation) by adding the same number to both sides of the equation.
  2. We can obtain an equivalent equation by subtracting the same number from both sides of the equation.

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

Solving x + a = b For x

x + a = b The a is associated with x by addition . Undo the association x + a a = b a by subtracting a from b o t h sides . x + 0 = b a a a = 0 and 0 is the additive identity . x + 0 = x . x = b a This equation is equivalent to the first equation, and it is solved for x .

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Solving x a = b For x

x a = b The a is associated with x by subtraction . Undo the association x a + a = b + a by adding a to b o t h sides . x + 0 = b + a a + a = 0 and 0 is the additive identity . x + 0 = x . x = b + a This equation is equivalent to the first equation, and it is solved for x .

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Method for solving x + a = b And x a = b For x

To solve the equation x + a = b for x , subtract a from both sides of the equation.
To solve the equation x a = b for x , add a to both sides of the equation.

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Sample set b

Solve x + 7 = 10 for x .

x + 7 = 10 7 is associated with x by addition . Undo the association x + 7 7 = 10 7 by subtracting 7 from b o t h sides . x + 0 = 3 7 7 = 0 and 0 is the additive identity . x + 0 = x . x = 3 x is isolated, and the equation x = 3 is equivalent to the original equation x + 7 = 10. Therefore, these two equation have the same solution . The solution to x = 3 is clearly 3. Thus, the solution to x + 7 = 10 is also 3.

Check : Substitute 3 for x in the original equation. x + 7 = 10 3 + 7 = 10 Is this correct? 10 = 10 Yes, this is correct .

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Solve m 2 = 9 for m .

m 2 = 9 2 is associated with m by subtraction . Undo the association m 2 + 2 = 9 + 2 by adding 2 from b o t h sides . m + 0 = 7 2 + 2 = 0 and 0 is the additive identity . m + 0 = m . m = 7

Check : Substitute 7 for m in the original equation. m 2 = 9 7 2 = 9 Is this correct? 9 = 9 Yes, this is correct .

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Use a calculator to solve this equation. Solve y 2.181 = 16.915 for y .

y 2.181 = 16.915 y 2.181 + 2.181 = 16.915 + 2.181 y = 14.734

On the Calculator
Type 16.915 Press + / Press + Type 2.181 Press = Display reads: 14.734

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Solve y + m = s for y .

y + m = s m is associated with y by addition . Undo the association y + m m = s m by subtracting m from b o t h sides . y + 0 = s m m m = 0 and 0 is the additive identity . y + 0 = y . y = s m

Check : Substitute s m for y in the original equation. y + m = s s m + m = s Is this correct? s = s True Yes, this is correct .

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Solve k 3 h = 8 h + 5 for k .

k 3 h = 8 h + 5 3 h is associated with k by subtraction . Undo the association k 3 h + 3 h = 8 h + 5 + 3 h by adding 3 h to b o t h sides . k + 0 = 5 h + 5 3 h + 3 h = 0 and 0 is the additive identity . k + 0 = k . k = 5 h + 5

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Practice set b

Solve y 3 = 8 for y .

y = 11

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Solve x + 9 = 4 for x .

x = 13

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Solve m + 6 = 0 for m .

m = 6

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Solve g 7.2 = 1.3 for g .

g = 8.5

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solve f + 2 d = 5 d for f .

f = 3 d

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Solve x + 8 y = 2 y 1 for x .

x = 6 y 1

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Solve y + 4 x 1 = 5 x + 8 for y .

y = x + 9

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Exercises

For the following problems, classify each of the equations as an identity, contradiction, or conditional equation.

g + g + g + g = 4 g

identity

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For the following problems, determine which of the literal equations have been solved for a variable. Write "solved" or "not solved."

4 a = y 6

not solved

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

y + 6 = 11

y = 17

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g + 164 = 123

g = 287

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x + 17 = 426

x = 443

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y + 17.003 = 1.056

y = 18.059

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Solve n + m = 4 for n .

n = 4 m

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Solve P + 3 Q 8 = 0 for P .

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Solve a + b 3 c = d 2 f for b .

b = a + 3 c + d 2 f

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Solve x 3 y + 5 z + 1 = 2 y 7 z + 8 for x .

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Solve 4 a 2 b + c + 11 = 6 a 5 b for c .

c = 2 a 3 b 11

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Exercises for review

( [link] ) Simplify ( 4 x 5 y 2 ) 3 .

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( [link] ) Write 20 x 3 y 7 5 x 5 y 3 so that only positive exponents appear.

4 y 4 x 2

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( [link] ) Write the number of terms that appear in the expression 5 x 2 + 2 x 6 + ( a + b ) , and then list them.

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( [link] ) Find the product. ( 3 x 1 ) 2 .

9 x 2 6 x + 1

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( [link] ) Specify the domain of the equation y = 5 x 2 .

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Questions & Answers

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
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
how to synthesize TiO2 nanoparticles by chemical methods
Zubear
what's the program
Jordan
?
Jordan
what chemical
Jordan
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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