<< Chapter < Page Chapter >> Page >

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

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
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Contemporary math applications' conversation and receive update notifications?

Ask