8.3 Solve equations with variables and constants on both sides  (Page 4/5)

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Solve: $5\left(x+3\right)=35.$

x = 4

Solve: $6\left(y-4\right)=-18.$

y = 1

Solve: $-\left(x+5\right)=7.$

Solution

 Simplify each side of the equation as much as possible by distributing. The only $x$ term is on the left side, so all variable terms are on the left side of the equation. Add 5 to both sides to get all constant terms on the right side of the equation. Simplify. Make the coefficient of the variable term equal to 1 by multiplying both sides by -1. Simplify. Check: Let $x=-12$ .

Solve: $-\left(y+8\right)=-2.$

y = −6

Solve: $-\left(z+4\right)=-12.$

z = 8

Solve: $4\left(x-2\right)+5=-3.$

Solution

 Simplify each side of the equation as much as possible. Distribute. Combine like terms The only $x$ is on the left side, so all variable terms are on one side of the equation. Add 3 to both sides to get all constant terms on the other side of the equation. Simplify. Make the coefficient of the variable term equal to 1 by dividing both sides by 4. Simplify. Check: Let $x=0$ .

Solve: $2\left(a-4\right)+3=-1.$

a = 2

Solve: $7\left(n-3\right)-8=-15.$

n = 2

Solve: $8-2\left(3y+5\right)=0.$

Solution

Be careful when distributing the negative.

 Simplify—use the Distributive Property. Combine like terms. Add 2 to both sides to collect constants on the right. Simplify. Divide both sides by −6. Simplify. Check: Let $y=-\frac{1}{3}$ .

Solve: $12-3\left(4j+3\right)=-17.$

$j=\frac{5}{3}$

Solve: $-6-8\left(k-2\right)=-10.$

$k=\frac{5}{2}$

Solve: $3\left(x-2\right)-5=4\left(2x+1\right)+5.$

Solution

 Distribute. Combine like terms. Subtract $3x$ to get all the variables on the right since $8>3$ . Simplify. Subtract 9 to get the constants on the left. Simplify. Divide by 5. Simplify. Check: Substitute: $-4=x$ .

Solve: $6\left(p-3\right)-7=5\left(4p+3\right)-12.$

p = −2

Solve: $8\left(q+1\right)-5=3\left(2q-4\right)-1.$

q = −8

Solve: $\frac{1}{2}\left(6x-2\right)=5-x.$

Solution

 Distribute. Add $x$ to get all the variables on the left. Simplify. Add 1 to get constants on the right. Simplify. Divide by 4. Simplify. Check: Let $x=\frac{3}{2}$ .

Solve: $\frac{1}{3}\left(6u+3\right)=7-u.$

u = 2

Solve: $\frac{2}{3}\left(9x-12\right)=8+2x.$

x = 4

In many applications, we will have to solve equations with decimals. The same general strategy will work for these equations.

Solve: $0.24\left(100x+5\right)=0.4\left(30x+15\right).$

Solution

 Distribute. Subtract $12x$ to get all the $x$ s to the left. Simplify. Subtract 1.2 to get the constants to the right. Simplify. Divide. Simplify. Check: Let $x=0.4$ .

Solve: $0.55\left(100n+8\right)=0.6\left(85n+14\right).$

1

Solve: $0.15\left(40m-120\right)=0.5\left(60m+12\right).$

−1

Key concepts

• Solve an equation with variables and constants on both sides
1. Choose one side to be the variable side and then the other will be the constant side.
2. Collect the variable terms to the variable side, using the Addition or Subtraction Property of Equality.
3. Collect the constants to the other side, using the Addition or Subtraction Property of Equality.
4. Make the coefficient of the variable 1, using the Multiplication or Division Property of Equality.
5. Check the solution by substituting into the original equation.
• General strategy for solving linear equations
1. Simplify each side of the equation as much as possible. Use the Distributive Property to remove any parentheses. Combine like terms.
2. Collect all the variable terms to one side of the equation. Use the Addition or Subtraction Property of Equality.
3. Collect all the constant terms to the other side of the equation. Use the Addition or Subtraction Property of Equality.
4. Make the coefficient of the variable term to equal to 1. Use the Multiplication or Division Property of Equality. State the solution to the equation.
5. Check the solution. Substitute the solution into the original equation to make sure the result is a true statement.

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
Jeannette has $5 and$10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
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How do i figure this problem out.
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Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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