# 0.1 Solving linear equations and inequalities: solving equations

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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. In this chapter, the emphasis is on the mechanics of equation solving, which clearly explains how to isolate a variable. The goal is to help the student feel more comfortable with solving applied problems. Ample opportunity is provided for the student to practice translating words to symbols, which is an important part of the "Five-Step Method" of solving applied problems (discussed in modules (<link document="m21980"/>) and (<link document="m21979"/>)). Objectives of this module: be able to identify various types of equations, understand the meaning of solutions and equivalent equations, be able to solve equations of the form x + a = b and x - a = b, be familiar with and able to solve literal equations.

## Overview

• Types of Equations
• Solutions and Equivalent Equations
• Literal Equations
• Solving Equations of the Form $x+a=b$ and $x-a=b$

## Identity

Some equations are always true. These equations are called identities. Identities are equations that are true for all acceptable values of the variable, that is, for all values in the domain of the equation.

$5x=5x$ is true for all acceptable values of $x$ .
$y+1=y+1$ is true for all acceptable values of $y$ .
$2+5=7$ is true, and no substitutions are necessary.

Some equations are never true. These equations are called contradictions. Contradictions are equations that are never true regardless of the value substituted for the variable.

$x=x+1$ is never true for any acceptable value of $x$ .
$0\text{\hspace{0.17em}}·\text{\hspace{0.17em}}k=14$ is never true for any acceptable value of $k$ .
$2=1$ is never true.

## Conditional equation

The truth of some equations is conditional upon the value chosen for the variable. Such equations are called conditional equations. Conditional equations are equations that are true for at least one replacement of the variable and false for at least one replacement of the variable.

$x+6=11$ is true only on the condition that $x=5$ .
$y-7=-1$ is true only on the condition that $y=6$ .

## Solutions and solving an equation

The collection of values that make an equation true are called solutions of the equation. An equation is solved when all its solutions have been found.

## Equivalent equations

Some equations have precisely the same collection of solutions. Such equations are called equivalent equations . The equations
$\begin{array}{cccc}2x+1=7,& 2x=6& \text{and}& x=3\end{array}$
are equivalent equations because the only value that makes each one true is 3.

## Sample set a

Tell why each equation is an identity, a contradiction, or conditional.

The equation $x-4=6$ is a conditional equation since it will be true only on the condition that $x=10$ .

The equation $x-2=x-2$ is an identity since it is true for all values of $x$ . For example,

$\begin{array}{cccccc}\text{if}\text{\hspace{0.17em}}x& =& 5,& 5-2& =& 5-2\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{true}\\ x& =& -7,& -7-2& =& -7-2\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{true}\end{array}$

The equation $a+5=a+1$ is a contradiction since every value of $a$ produces a false statement. For example,

$\begin{array}{cccccc}\text{if}\text{\hspace{0.17em}}a& =& 8,& 8+5& =& 8+1\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{false}\\ \text{if}\text{\hspace{0.17em}}a& =& -2,& -2+5& =& -2+1\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{false}\end{array}$

## Practice set a

For each of the following equations, write "identity," "contradiction," or "conditional." If you can, find the solution by making an educated guess based on your knowledge of arithmetic.

#### Questions & Answers

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
characteristics of micro business
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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