# 3.2 Solving linear equations by combining properties

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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.

what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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
what is Nano technology ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
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
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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