# Elementary algebra: solving linear equations in one variable  (Page 3/3)

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Solve for a: $-4a+2-a=3+5a-2$ The solution set is $\left\{\frac{1}{10}\right\}$

## 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: $5\left(3x+2\right)-2=-2\left(1-7x\right)$ The solution set is $\left\{-10\right\}$ .

## 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, $\Re$ .
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: $4\left(x+5\right)+6=2\left(2x+3\right)$ $\begin{array}{cccc}4\left(x+5\right)+6& =& 2\left(2x+3\right)& \mathit{\text{Distribute}}\hfill \\ 4x{+}{20}{+}{6}& =& 4x+6& \mathit{\text{Add same side like terms}}\text{.}\hfill \\ 4x+26& =& 4x+6& \\ 4x+26{-}{4}{x}& =& 4x+6{-}{4}{x}& \mathit{\text{Subtract 4x on both sides.}}\hfill \\ 26& =& 6& \mathit{\text{False}}\hfill \end{array}$ There is no solution, $\varnothing$ .

Solve for y: $3\left(3y+5\right)+5=10\left(y+2\right)-y$ The equation is an identity, the solution set consists of all real numbers, $\Re$ .

## 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: $P=2a+b$ Solution: $a=\frac{P-b}{2}$

Solve for x: $z=\frac{x+y}{2}$ Solution $x=2z-y$

Yes

No

Yes

Yes

No

## Solving in one step

$x=-3$

$y=-5$

$x=\frac{5}{6}$

$x=\frac{5}{6}$

$x=-11$

$a=10$

$y=3$

$x=-\frac{3}{2}$

$t=-3$

$x=-\frac{3}{14}$

## Solve in two steps

$a=10$

$y=5$

$x=0$

$x=\frac{2}{15}$

$y=7$

$x=-\frac{5}{2}$

$a=\frac{1}{8}$

$x=1$

$y=\frac{1}{3}$

$x=-3$

## Solve in multiple steps

$x=-12$

$y=-4$

$a=-\frac{26}{15}$

$x=-1$

No Solution, $\varnothing$

All Reals, $\Re$

$y=\frac{1}{2}$

$x=2$

$a=-\frac{10}{9}$

All Reals, $\Re$

## Literal equations

$w=\frac{P-2l}{2}$

$b=P-a-c$

$C=\frac{5F-160}{9}$

$r=\frac{C}{2\pi }$

$y=-5z+x$

do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
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
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?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
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.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
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