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Solve: 8 x + 9 x 5 x = −3 + 15 .

Solution

Start by combining like terms to simplify each side.

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Combine like terms. .
Divide both sides by 12 to isolate x. .
Simplify. .
Check your answer. Let x = 1
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Solve: 7 x + 6 x 4 x = −8 + 26 .

x = 2

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Solve: 11 n 3 n 6 n = 7 17 .

n = −5

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Solve: 11 20 = 17 y 8 y 6 y .

Solution

Simplify each side by combining like terms.

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Simplify each side. .
Divide both sides by 3 to isolate y. .
Simplify. .
Check your answer. Let y = −3
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Notice that the variable ended up on the right side of the equal sign when we solved the equation. You may prefer to take one more step to write the solution with the variable on the left side of the equal sign.

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Solve: 18 27 = 15 c 9 c 3 c .

c = −3

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Solve: 18 22 = 12 x x 4 x .

x = 4 7

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Solve: −3 ( n 2 ) 6 = 21 .

Solution

Remember—always simplify each side first.

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Distribute. .
Simplify. .
Divide both sides by -3 to isolate n. .
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Check your answer. Let n = −7 .
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Solve: −4 ( n 2 ) 8 = 24 .

n = −6

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Solve: −6 ( n 2 ) 12 = 30 .

n = −5

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Key concepts

  • Division and Multiplication Properties of Equality
    • Division Property of Equality: For all real numbers a, b, c, and c 0 , if a = b , then a c = b c .
    • Multiplication Property of Equality: For all real numbers a, b, c, if a = b , then a c = b c .

Practice makes perfect

Solve Equations Using the Division and Multiplication Properties of Equality

In the following exercises, solve each equation for the variable using the Division Property of Equality and check the solution.

In the following exercises, solve each equation for the variable using the Multiplication Property of Equality and check the solution.

Solve Equations That Need to be Simplified

In the following exercises, solve the equation.

8 a + 3 a 6 a = −17 + 27

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6 y 3 y + 12 y = −43 + 28

y = −1

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−9 x 9 x + 2 x = 50 2

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−5 m + 7 m 8 m = −6 + 36

m = −5

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100 16 = 4 p 10 p p

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−18 7 = 5 t 9 t 6 t

t = 5 2

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5 12 q + 1 2 q = 25 3

q = 24

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0.25 d + 0.10 d = 6 0.75

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0.05 p 0.01 p = 2 + 0.24

p = 56

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Everyday math

Balloons Ramona bought 18 balloons for a party. She wants to make 3 equal bunches. Find the number of balloons in each bunch, b , by solving the equation 3 b = 18 .

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Teaching Connie’s kindergarten class has 24 children. She wants them to get into 4 equal groups. Find the number of children in each group, g , by solving the equation 4 g = 24 .

6 children

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Ticket price Daria paid $36.25 for 5 children’s tickets at the ice skating rink. Find the price of each ticket, p , by solving the equation 5 p = 36.25 .

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Unit price Nishant paid $12.96 for a pack of 12 juice bottles. Find the price of each bottle, b , by solving the equation 12 b = 12.96 .

$1.08

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Fuel economy Tania’s SUV gets half as many miles per gallon (mpg) as her husband’s hybrid car. The SUV gets 18 mpg . Find the miles per gallons, m , of the hybrid car, by solving the equation 1 2 m = 18 .

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Fabric The drill team used 14 yards of fabric to make flags for one-third of the members. Find how much fabric, f , they would need to make flags for the whole team by solving the equation 1 3 f = 14 .

42 yards

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Writing exercises

Frida started to solve the equation −3 x = 36 by adding 3 to both sides. Explain why Frida’s method will result in the correct solution.

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Emiliano thinks x = 40 is the solution to the equation 1 2 x = 80 . Explain why he is wrong.

Answer will vary.

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Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

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After reviewing this checklist, what will you do to become confident for all objectives?

Questions & Answers

what is the stm
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
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?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
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?
s. Reply
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
Devang Reply
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
Smarajit Reply
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A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
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why surface tension is zero at critical temperature
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
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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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