# 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, $\Re$ .
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: $4\left(x+5\right)+6=2\left(2x+3\right)$ $\begin{array}{cccc}4\left(x+5\right)+6& =& 2\left(2x+3\right)& \mathit{\text{Distribute}}\hfill \\ 4x{+}{20}{+}{6}& =& 4x+6& \mathit{\text{Add same side like terms}}\text{.}\hfill \\ 4x+26& =& 4x+6& \\ 4x+26{-}{4}{x}& =& 4x+6{-}{4}{x}& \mathit{\text{Subtract 4x on both sides.}}\hfill \\ 26& =& 6& \mathit{\text{False}}\hfill \end{array}$ 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

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

$b=P-a-c$

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

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

$y=-5z+x$

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
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?
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research.net
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sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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there is no specific books for beginners but there is book called principle of nanotechnology
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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
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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.
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s.
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for screen printed electrodes ?
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What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
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in general
s.
Graphene has a hexagonal structure
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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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