# 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.

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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
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
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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