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x + 1 = 10

conditional, x = 9

y 4 = 7

conditional, y = 11

5 a = 25

conditional, a = 5

x 4 = 9

conditional, x = 36

18 b = 6

conditional, b = 3

y 2 = y 2

identity

x + 4 = x 3

contradiction

x + x + x = 3 x

identity

8 x = 0

conditional, x = 0

m 7 = 5

conditional, m = 2

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 .

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 .

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.

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 .

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 .

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

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 .

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

Practice set b

Solve y 3 = 8 for y .

y = 11

Solve x + 9 = 4 for x .

x = 13

Solve m + 6 = 0 for m .

m = 6

Solve g 7.2 = 1.3 for g .

g = 8.5

solve f + 2 d = 5 d for f .

f = 3 d

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

x = 6 y 1

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

y = x + 9

Exercises

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

m + 6 = 15

conditional

y 8 = 12

x + 1 = x + 1

identity

k 2 = k 3

g + g + g + g = 4 g

identity

x + 1 = 0

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

y = 3 x + 7

solved

m = 2 k + n 1

4 a = y 6

not solved

h k = 2 k + h

2 a = a + 1

not solved

5 m = 2 m 7

m = m

not solved

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

h 8 = 14

k + 10 = 1

k = 9

m 2 = 5

y + 6 = 11

y = 17

y 8 = 1

x + 14 = 0

x = 14

m 12 = 0

g + 164 = 123

g = 287

h 265 = 547

x + 17 = 426

x = 443

h 4.82 = 3.56

y + 17.003 = 1.056

y = 18.059

k + 1.0135 = - 6.0032

Solve n + m = 4 for n .

n = 4 m

Solve P + 3 Q 8 = 0 for P .

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

b = a + 3 c + d 2 f

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

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

c = 2 a 3 b 11

Exercises for review

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

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

4 y 4 x 2

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

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

9 x 2 6 x + 1

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

Questions & Answers

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
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
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 did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Linear equations. OpenStax CNX. Jun 15, 2015 Download for free at https://legacy.cnx.org/content/col11828/1.1
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