# 5.3 Further techniques in equation solving

 Page 1 / 2
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 comfortable with combining techniques in equation solving, be able to recognize identities and contradictions.

## Overview

• Combining Techniques in Equation Solving
• Recognizing Identities and Contrdictions

## Combining techniques in equation solving

In Sections [link] and [link] we worked with techniques that involved the use of addition, subtraction, multiplication, and division to solve equations. We can combine these techniques to solve more complicated equations. To do so, it is helpful to recall that an equation is solved for a particular variable when all other numbers and/or letters have been disassociated from it and it is alone on one side of the equal sign. We will also note that

To associate numbers and letters we use the order of operations.

1. Multiply/divide

To undo an association between numbers and letters we use the order of operations in reverse.

2. Multiply/divide

## Sample set a

Solve $4x-7=9$ for $x.$

$\begin{array}{llll}\hfill 4x-7& =\hfill & 9\hfill & \text{First,}\text{\hspace{0.17em}}\text{undo}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{association}\text{\hspace{0.17em}}\text{between}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}7.\hfill \\ \hfill & \hfill & \hfill & \text{\hspace{0.17em}}\text{The}\text{\hspace{0.17em}}7\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{associated}\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{subtraction}\text{.}\hfill \\ \hfill & \hfill & \hfill & \text{Undo}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{association}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{adding}\text{\hspace{0.17em}}7\text{\hspace{0.17em}}\text{to}\text{\hspace{0.17em}}both\text{\hspace{0.17em}}\text{sides}\text{.}\hfill \\ \hfill 4x-7+7& =\hfill & 9+7\hfill & \hfill \\ \hfill 4x& =\hfill & 16\hfill & \text{Now,}\text{\hspace{0.17em}}\text{undo}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{association}\text{\hspace{0.17em}}\text{between}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}4.\hfill \\ \hfill & \hfill & \hfill & \text{\hspace{0.17em}}\text{The}\text{\hspace{0.17em}}4\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{associated}\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{multiplication}.\hfill \\ \hfill & \hfill & \hfill & \text{Undo}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{association}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{dividing}\text{\hspace{0.17em}}both\text{\hspace{0.17em}}\text{sides}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}4.\hfill \\ \hfill \frac{4x}{4}& =& \frac{16}{4}& \\ \hfill 16-7& =\hfill & 9\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill x& =\hfill & 4\hfill & \hfill \end{array}$

$\begin{array}{lllll}Check:\hfill & \hfill 4\left(4\right)-7& =\hfill & 9\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill 9& =\hfill & 9\hfill & \text{Yes,}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{correct}\text{.}\hfill \end{array}$

Solve $\frac{3y}{4}-5=-11.$

$\begin{array}{llll}\hfill \frac{3y}{4}-5& =\hfill & -11\hfill & -5\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{associated}\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}y\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{subtraction}\text{.}\hfill \\ \hfill & \hfill & \hfill & \text{\hspace{0.17em}}\text{Undo}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{association}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{adding}\text{\hspace{0.17em}}5\text{\hspace{0.17em}}\text{to}\text{\hspace{0.17em}}both\text{\hspace{0.17em}}\text{sides}\text{.}\hfill \\ \hfill \frac{3y}{4}-5+5& =\hfill & -11+5\hfill & \hfill \\ \hfill \frac{3y}{4}& =\hfill & -6\hfill & 4\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{associated}\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}y\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{division}\text{.}\hfill \\ \hfill & \hfill & \hfill & \text{\hspace{0.17em}}\text{Undo}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{association}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{multiplying}\text{\hspace{0.17em}}both\text{\hspace{0.17em}}\text{sides}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}4.\hfill \\ \hfill 4\cdot \frac{3y}{4}& =\hfill & 4\left(-6\right)\hfill & \hfill \\ \hfill 4\cdot \frac{3y}{4}& =\hfill & 4\left(-6\right)\hfill & \hfill \\ \hfill 3y& =\hfill & -24\hfill & 3\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{associated}\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}y\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{multiplication}\text{.}\hfill \\ \hfill & \hfill & \hfill & \text{\hspace{0.17em}}\text{Undo}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{association}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{dividing}\text{\hspace{0.17em}}both\text{\hspace{0.17em}}\text{sides}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}3.\hfill \\ \hfill \frac{3y}{3}& =& \frac{-24}{3}& \hfill \\ \hfill \frac{3y}{3}& =\hfill & -8\hfill & \hfill \\ \hfill y& =\hfill & -8\hfill & \hfill \end{array}$

$\begin{array}{lllll}Check:\hfill & \hfill \frac{3\left(-8\right)}{4}-5& =\hfill & -11\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill \frac{-24}{4}-5& =\hfill & -11\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill -6-5& =\hfill & -11\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill -11& =\hfill & -11\hfill & \text{Yes,}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{correct}\text{.}\hfill \end{array}$

Solve $\frac{8a}{3b}+2m=6m-5$ for $a.$

$\begin{array}{llll}\hfill \frac{8a}{3b}+2m& =\hfill & 6m-5\hfill & 2m\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{associated}\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}a\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{addition}\text{.}\text{\hspace{0.17em}}\text{Undo}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{association}\hfill \\ \hfill & \hfill & \hfill & \text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{subtracting}\text{\hspace{0.17em}}2m\text{\hspace{0.17em}}\text{from}\text{\hspace{0.17em}}both\text{\hspace{0.17em}}\text{sides}.\hfill \\ \hfill \frac{8a}{3b}+2m-2m& =\hfill & 6m-5-2m\hfill & \hfill \\ \hfill \frac{8a}{3b}& =\hfill & 4m-5\hfill & 3b\text{\hspace{0.17em}}\text{\hspace{0.17em}}associated\text{\hspace{0.17em}}with\text{\hspace{0.17em}}a\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{division}\text{.}\text{\hspace{0.17em}}\text{Undo}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{association}\hfill \\ \hfill & \hfill & \hfill & \text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{multiplying}both\text{\hspace{0.17em}}\text{sides}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}3b.\hfill \\ \hfill \left(3b\right)\left(\frac{8a}{3b}\right)& =\hfill & 3b\left(4m-5\right)\hfill & \hfill \\ \hfill 8a& =\hfill & 12bm-15b\hfill & 8\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{associated}\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}a\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{multiplication}\text{.}\text{\hspace{0.17em}}\text{Undo}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\hfill \\ \hfill & \hfill & \hfill & \text{multiplication}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}\text{dividing}both\text{\hspace{0.17em}}\text{sides}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}8.\hfill \\ \hfill \frac{8a}{8}& =\hfill & \frac{12bm-15b}{8}\hfill & \hfill \\ \hfill a& =\hfill & \frac{12bm-15b}{8}\hfill & \hfill \end{array}$

## Practice set a

Solve $3y-1=11$ for $y.$

$y=4$

Solve $\frac{5m}{2}+6=1$ for $m.$

$m=-2$

Solve $2n+3m=4$ for $n.$

$n=\frac{4-3m}{2}$

Solve $\frac{9k}{2h}+5=p-2$ for $k.$

$k=\frac{2hp-14h}{9}$

Sometimes when solving an equation it is necessary to simplify the expressions composing it.

## Sample set b

Solve $4x+1-3x=\left(-2\right)\left(4\right)$ for $x.$

$\begin{array}{lll}\hfill 4x+1-3x& =\hfill & \left(-2\right)\left(4\right)\hfill \\ \hfill x+1& =\hfill & -8\hfill \\ \hfill x& =\hfill & -9\hfill \end{array}$

$\begin{array}{lllll}Check:\hfill & \hfill 4\left(-9\right)+1-3\left(-9\right)& =\hfill & -8\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill -36+1+27& =\hfill & -8\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill -8& =\hfill & -8\hfill & \text{Yes,}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{correct}\text{.}\hfill \end{array}$

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
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
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
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?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Please keep in mind that it's not allowed to promote any social groups (whatsapp, facebook, etc...), exchange phone numbers, email addresses or ask for personal information on QuizOver's platform.