5.3 Further techniques in equation solving  (Page 2/2)

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Solve $3\left(m-6\right)-2m=-4+1$ for $m.$

$\begin{array}{lll}\hfill 3\left(m-6\right)-2m& =\hfill & -4+1\hfill \\ \hfill 3m-18-2m& =\hfill & -3\hfill \\ \hfill m-18& =\hfill & -3\hfill \\ \hfill m& =\hfill & 15\hfill \end{array}$

$\begin{array}{lllll}Check:\hfill & \hfill 3\left(15-6\right)-2\left(15\right)& =\hfill & -4+1\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill 3\left(9\right)-30& =\hfill & -3\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill 27-30& =\hfill & -3\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill -3& =\hfill & -3\hfill & \text{Yes,}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{correct}\text{.}\hfill \end{array}$

Practice set b

Solve and check each equation.

$16x-3-15x=8$ for $x.$

$x=11$

$4\left(y-5\right)-3y=-1$ for $y.$

$y=19$

$-2\left({a}^{2}+3a-1\right)+2{a}^{2}+7a=0$ for $a.$

$a=-2$

$5m\left(m-2a-1\right)-5{m}^{2}+2a\left(5m+3\right)=10$ for $a.$

$a=\frac{10+5m}{6}$

Often the variable we wish to solve for will appear on both sides of the equal sign. We can isolate the variable on either the left or right side of the equation by using the techniques of Sections [link] and [link] .

Sample set c

Solve $6x-4=2x+8$ for $x.$

$\begin{array}{llll}\hfill 6x-4& =\hfill & 2x+8\hfill & \text{To}\text{\hspace{0.17em}}\text{isolate}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\text{on}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{left}\text{\hspace{0.17em}}\text{side,}\text{\hspace{0.17em}}\text{subtract}\text{\hspace{0.17em}}2m\text{\hspace{0.17em}}\text{from}\text{\hspace{0.17em}}\text{both}\text{\hspace{0.17em}}\text{sides}.\hfill \\ \hfill 6x-4-2x& =\hfill & 2x+8-2x\hfill & \hfill \\ \hfill 4x-4& =\hfill & 8\hfill & \text{Add}\text{\hspace{0.17em}}4\text{\hspace{0.17em}}\text{to}\text{\hspace{0.17em}}\text{both}\text{\hspace{0.17em}}\text{sides}.\hfill \\ \hfill 4x-4+4& =\hfill & 8+4\hfill & \hfill \\ \hfill 4x& =& 12\hfill & \text{Divide}\text{\hspace{0.17em}}\text{both}\text{\hspace{0.17em}}\text{sides}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}4.\hfill \\ \hfill \frac{4x}{4}& =\hfill & \frac{12}{4}\hfill & \\ \hfill x& =\hfill & 3\hfill & \end{array}$

$\begin{array}{lllll}Check:\hfill & \hfill 6\left(3\right)-4& =\hfill & 2\left(3\right)+8\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill 18-4& =\hfill & 6+8\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill 14& =\hfill & 14\hfill & \text{Yes,}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{correct}\text{.}\hfill \end{array}$

Solve $6\left(1-3x\right)+1=2x-\left[3\left(x-7\right)-20\right]$ for $x.$

$\begin{array}{llll}\hfill 6-18x+1& =\hfill & 2x-\left[3x-21-20\right]\hfill & \hfill \\ \hfill -18x+7& =\hfill & 2x-\left[3x-41\right]\hfill & \\ \hfill -18x+7& =\hfill & 2x-3x+41\hfill & \\ \hfill -18x+7& =\hfill & -x+41\hfill & \text{To}\text{\hspace{0.17em}}\text{isolate}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\text{on}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{right}\text{\hspace{0.17em}}\text{side,}\text{\hspace{0.17em}}\text{add}\text{\hspace{0.17em}}18x\text{\hspace{0.17em}}\text{to}\text{\hspace{0.17em}}\text{both}\text{\hspace{0.17em}}\text{sides}.\hfill \\ \hfill -18x+7+18x& =\hfill & -x+41+18x\hfill & \hfill \\ \hfill 7& =\hfill & 17x+41\hfill & \text{Subtract}\text{\hspace{0.17em}}41\text{\hspace{0.17em}}\text{from}\text{\hspace{0.17em}}\text{both}\text{\hspace{0.17em}}\text{sides}.\hfill \\ \hfill 7-41& =\hfill & 17x+41-41\hfill & \hfill \\ \hfill -34& =\hfill & 17x\hfill & \text{Divide}\text{\hspace{0.17em}}\text{both}\text{\hspace{0.17em}}\text{sides}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}17.\hfill \\ \hfill \frac{-34}{17}& =\hfill & \frac{17x}{17}\hfill & \\ \hfill -2& =\hfill & x\hfill & \text{Since}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{equation}\text{\hspace{0.17em}}-2=x\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{equivalent}\text{\hspace{0.17em}}\text{to}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{equation}\hfill \\ \hfill & \hfill & \hfill & x=-2,\text{\hspace{0.17em}}\text{we}\text{\hspace{0.17em}}\text{can}\text{\hspace{0.17em}}\text{write}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{answer}\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}x=-2.\hfill \\ \hfill x& =\hfill & -2\hfill & \end{array}$

$\begin{array}{lllll}Check:\hfill & \hfill 6\left(1-3\left(-2\right)\right)+1& =\hfill & 2\left(-2\right)-\left[3\left(-2-7\right)-20\right]\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill 6\left(1+6\right)+1& =\hfill & -4-\left[3\left(-9\right)-20\right]\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill 6\left(7\right)+1& =\hfill & -4-\left[-27-20\right]\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill 42+1& =\hfill & -4-\left[-47\right]\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill 43& =\hfill & -4+47\hfill & \text{Is}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{correct?}\hfill \\ \hfill & \hfill 43& =\hfill & 43\hfill & \text{Yes,}\text{\hspace{0.17em}}\text{this}\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{correct}\text{.}\hfill \end{array}$

Practice set c

Solve $8a+5=3a-5$ for $a.$

$a=-2$

Solve $9y+3\left(y+6\right)=15y+21$ for $y.$

$y=-1$

Solve $3k+2\left[4\left(k-1\right)+3\right]=63-2k$ for $k.$

$k=5$

As we noted in Section [link] , some equations are identities and some are contradictions. As the problems of Sample Set D will suggest,

Recognizing an identity

1. If, when solving an equation, all the variables are eliminated and a true statement results, the equation is an identity.

1. If, when solving an equation, all the variables are eliminated and a false statement results, the equation is a contradiction.

Sample set d

Solve $9x+3\left(4-3x\right)=12$ for $x.$

$\begin{array}{lll}\hfill 9x+12-9x& =\hfill & 12\hfill \\ \hfill 12& =\hfill & 12\hfill \end{array}$

The variable has been eliminated and the result is a true statement. The original equation is an identity.

Solve $-2\left(10-2y\right)-4y+1=-18$ for $y.$

$\begin{array}{lll}\hfill -20+4y-4y+1& =\hfill & -18\hfill \\ \hfill -19& =\hfill & -18\hfill \end{array}$

The variable has been eliminated and the result is a false statement. The original equation is a contradiction.

Practice set d

Classify each equation as an identity or a contradiction.

$6x+3\left(1-2x\right)=3$

identity, $3=3$

$-8m+4\left(2m-7\right)=28$

contradiction, $-28=28$

$3\left(2x-4\right)-2\left(3x+1\right)+14=0$

identity, $0=0$

$-5\left(x+6\right)+8=3\left[4-\left(x+2\right)\right]-2x$

contradiction, $-22=6$

Exercises

For the following problems, solve each conditional equation. If the equation is not conditional, identify it as an identity or a contradiction.

$3x+1=16$

$x=5$

$6y-4=20$

$4a-1=27$

$a=7$

$3x+4=40$

$2y+7=-3$

$y=-5$

$8k-7=-23$

$5x+6=-9$

$x=-3$

$7a+2=-26$

$10y-3=-23$

$y=-2$

$14x+1=-55$

$\frac{x}{9}+2=6$

$x=36$

$\frac{m}{7}-8=-11$

$\frac{y}{4}+6=12$

$y=24$

$\frac{x}{8}-2=5$

$\frac{m}{11}-15=-19$

$m=-44$

$\frac{k}{15}+20=10$

$6+\frac{k}{5}=5$

$k=-5$

$1-\frac{n}{2}=6$

$\frac{7x}{4}+6=-8$

$x=-8$

$\frac{-6m}{5}+11=-13$

$\frac{3k}{14}+25=22$

$k=-14$

$3\left(x-6\right)+5=-25$

$16\left(y-1\right)+11=-85$

$y=-5$

$6x+14=5x-12$

$23y-19=22y+1$

$y=20$

$-3m+1=3m-5$

$8k+7=2k+1$

$k=-1$

$12n+5=5n-16$

$2\left(x-7\right)=2x+5$

$-4\left(5y+3\right)+5\left(1+4y\right)=0$

$3x+7=-3-\left(x+2\right)$

$x=-3$

$4\left(4y+2\right)=3y+2\left[1-3\left(1-2y\right)\right]$

$5\left(3x-8\right)+11=2-2x+3\left(x-4\right)$

$x=\frac{19}{14}$

$12-\left(m-2\right)=2m+3m-2m+3\left(5-3m\right)$

$-4\cdot k-\left(-4-3k\right)=-3k-2k-\left(3-6k\right)+1$

$k=3$

$3\left[4-2\left(y+2\right)\right]=2y-4\left[1+2\left(1+y\right)\right]$

$-5\left[2m-\left(3m-1\right)\right]=4m-3m+2\left(5-2m\right)+1$

$m=2$

For the following problems, solve the literal equations for the indicated variable. When directed, find the value of that variable for the given values of the other variables.

Solve $I=\frac{E}{R}$ for $R.$ Find the value of $R$ when $I=0.005$ and $E=0.0035.$

Solve $P=R-C$ for $R.$ Find the value of $R$ when $P=27$ and $C=85.$

$R=112$

Solve $z=\frac{x-\overline{x}}{s}$ for $x.$ Find the value of $x$ when $z=1.96,$ $s=2.5,$ and $\overline{x}=15.$

Solve $F=\frac{{S}_{x}^{2}}{{S}_{y}^{2}}$ for ${S}_{x}^{2}\cdot {S}_{x}^{2}$ represents a single quantity. Find the value of ${S}_{x}^{2}$ when $F=2.21$ and ${S}_{y}^{2}=3.24.$

${S}_{x}{}^{2}=F·{S}_{y}{}^{2};\text{\hspace{0.17em}}{S}_{x}{}^{2}=7.1604$

Solve $p=\frac{nRT}{V}$ for $R.$

Solve $x=4y+7$ for $y.$

$y=\frac{x-7}{4}$

Solve $y=10x+16$ for $x.$

Solve $2x+5y=12$ for $y.$

$y=\frac{-2x+12}{5}$

Solve $-9x+3y+15=0$ for $y.$

Solve $m=\frac{2n-h}{5}$ for $n.$

$n=\frac{5m+h}{2}$

Solve $t=\frac{Q+6P}{8}$ for $P.$

Solve $=\frac{\square \text{\hspace{0.17em}}+9j}{\Delta }$ for $j$ .

Solve for .

Exercises for review

( [link] ) Simplify ${\left(x+3\right)}^{2}{\left(x-2\right)}^{3}{\left(x-2\right)}^{4}\left(x+3\right).$

${\left(x+3\right)}^{3}{\left(x-2\right)}^{7}$

( [link] ) Find the product. $\left(x-7\right)\left(x+7\right).$

( [link] ) Find the product. ${\left(2x-1\right)}^{2}.$

$4{x}^{2}-4x+1$

( [link] ) Solve the equation $y-2=-2.$

( [link] ) Solve the equation $\frac{4x}{5}=-3.$

$x=\frac{-15}{4}$

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
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
can nanotechnology change the direction of the face of the world
how did you get the value of 2000N.What calculations are needed to arrive at it
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