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Solve for a: 4 a + 2 a = 3 + 5 a 2 4 a + 2 a = 3 + 5 a 2 Add same side like terms first. 5 a + 2 = 5 a + 1 5 a + 2 5 a = 5 a + 1 5 a Subtract 5a from both sides . 10 a + 2 = 1 10 a + 2 2 = 1 2 Subtract 2 from both sides . 10 a = 1 10 a 10 = 1 10 Divide both sides by -10 . a = 1 10 The solution set is { 1 10 }

Simplifying expressions first

When solving linear equations the goal is to determine what value, if any, will solve the equation. A general guideline is to use the order of operations to simplify the expressions on both sides first.

Solve for x: 5 ( 3 x + 2 ) 2 = 2 ( 1 7 x ) 5 ( 3 x + 2 ) 2 = 2 ( 1 7 x ) Distribute . 15 x + 10 2 = 2 + 14 x Add same side like terms. 15 x + 8 = 2 + 14 x 15 x + 8 14 x = 2 + 14 x 14 x Subtract 14x on both sides. x + 8 = 2 x + 8 8 = 2 8 Subtract 8 on both sides. x = 10 The solution set is { 10 } .

Video Example 02

Conditional equations, identities, and contradictions

There are three different kinds of equations defined as follows.

Conditional Equation
A conditional equation is true for particular values of the variable.
Identity
An identity is an equation that is true for all possible values of the variable. For example, x = x has a solution set consisting of all real numbers, .
Contradiction
A contradiction is an equation that is never true and thus has no solutions. For example, x + 1 = x has no solution. No solution can be expressed as the empty set {    } = .

So far we have seen only conditional linear equations which had one value in the solution set. If when solving an equation and the end result is an identity, like say 0 = 0, then any value will solve the equation. If when solving an equation the end result is a contradiction, like say 0 = 1, then there is no solution.

Solve for x: 4 ( x + 5 ) + 6 = 2 ( 2 x + 3 ) 4 ( x + 5 ) + 6 = 2 ( 2 x + 3 ) Distribute 4 x + 20 + 6 = 4 x + 6 Add same side like terms . 4 x + 26 = 4 x + 6 4 x + 26 4 x = 4 x + 6 4 x Subtract 4x on both sides. 26 = 6 False There is no solution, .

Solve for y: 3 ( 3 y + 5 ) + 5 = 10 ( y + 2 ) y 3 ( 3 y + 5 ) + 5 = 10 ( y + 2 ) y Distribute 9 y + 15 + 5 = 10 y + 20 y Add same side like terms . 9 y + 20 = 9 y + 20 9 y + 20 20 = 9 y + 20 20 Subtract 20 on both sides . 9 y = 9 y 9 y 9 y = 9 y 9 y Subtract 9y on both sides . 0 = 0 True The equation is an identity, the solution set consists of all real numbers, .

Linear literal equations

Literal equations, or formulas, usually have more than one variable. Since the letters are placeholders for values, the steps for solving them are the same. Use the properties of equality to isolate the indicated variable.

Solve for a: P = 2 a + b P = 2 a + b P b = 2 a + b b Subtract b on both sides. P b = 2 a P b 2 = 2 a 2 Divide both sides by 2. P b 2 = a Solution: a = P b 2

Solve for x: z = x + y 2 z = x + y 2 2 z = 2 x + y 2 Multiply both sides by 2 . 2 z = x + y 2 z y = x + y y Subtract y on both sides . 2 z y = x Solution x = 2 z y

Exercises

Checking solutions

Is  x = 7  a solution to 3 x + 5 = 16 ?

Yes

Is  x = 2  a solution to  2 x 7 = 28 ?

No

Is  x = 3  a solution to  1 3 x 4 = 5 ?

Yes

Is  x = 2  a solution to  3 x 5 = 2 x 15 ?

Yes

Is  x = 1 2  a solution to  3 ( 2 x + 1 ) = 4 x 3 ?

No

Solving in one step

Solve for x:   x 5 = 8

x = 3

Solve for y:   4 + y = 9

y = 5

Solve for x:   x 1 2 = 1 3

x = 5 6

Solve for x:   x + 2 1 2 = 3 1 3

x = 5 6

Solve for x:   4 x = 44

x = 11

Solve for a:  3 a = 30

a = 10

Solve for y:   27 = 9 y

y = 3

Solve for x:   x 3 = 1 2

x = 3 2

Solve for t:   t 12 = 1 4

t = 3

Solve for x:   7 3 x = 1 2

x = 3 14

Solve in two steps

Solve for a:   3 a 7 = 23

a = 10

Solve for y:   3 y + 2 = 13

y = 5

Solve for x:   5 x + 8 = 8

x = 0

Solve for x:   1 2 x + 1 3 = 2 5

x = 2 15

Solve for y:   3 2 y = 11

y = 7

Solve for x:   10 = 2 x 5

x = 5 2

Solve for a:   4 a 2 3 = 1 6

a = 1 8

Solve for x:   3 5 x 1 2 = 1 10

x = 1

Solve for y:   4 5 y + 1 3 = 1 15

y = 1 3

Solve for x:   x 5 = 2

x = 3

Solve in multiple steps

Solve for x:   3 x 5 = 2 x 17

x = 12

Solve for y:   2 y 7 = 3 y + 13

y = 4

Solve for a:   1 2 a 2 3 = a + 1 5

a = 26 15

Solve for x:   2 + 4 x + 9 = 7 x + 8 2 x

x = 1

Solve for a:   3 a + 5 x = 2 a + 7

No Solution,

Solve for b:   7 b + 3 = 2 5 b + 1 2 b

All Reals,

Solve for y:   5 ( 2 y 3 ) + 2 = 12

y = 1 2

Solve for x:   3 2 ( x + 4 ) = 3 ( 4 x 5 )

x = 2

Solve for a:   3 ( 2 a 3 ) + 2 = 3 ( a + 7 )

a = 10 9

Solve for x:   10 ( 3 x + 5 ) 5 ( 4 x + 2 ) = 2 ( 5 x + 20 )

All Reals,

Literal equations

Solve for w:   P = 2 l + 2 w

w = P 2 l 2

Solve for b:   P = a + b + c

b = P a c

Solve for C:   F = 9 5 C + 32

C = 5 F 160 9

Solve for r:   C = 2 π r

r = C 2 π

Solve for y:   z = x y 5

y = 5 z + x

Questions & Answers

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 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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
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