# 8.2 Solve equations using the division and multiplication properties  (Page 2/2)

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Solve: $8x+9x-5x=-3+15.$

## Solution

Start by combining like terms to simplify each side.

 Combine like terms. Divide both sides by 12 to isolate x. Simplify. Check your answer. Let $x=1$

Solve: $7x+6x-4x=-8+26.$

x = 2

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

n = −5

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

## Solution

Simplify each side by combining like terms.

 Simplify each side. Divide both sides by 3 to isolate y. Simplify. Check your answer. Let $y=-3$

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.

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

c = −3

Solve: $18-22=12x-x-4x.$

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

Solve: $-3\left(n-2\right)-6=21.$

## Solution

Remember—always simplify each side first.

 Distribute. Simplify. Divide both sides by -3 to isolate n. Check your answer. Let $n=-7$ .

Solve: $-4\left(n-2\right)-8=24.$

n = −6

Solve: $-6\left(n-2\right)-12=30.$

n = −5

## Key concepts

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

## 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.

$8x=32$

$7p=63$

9

$-5c=55$

$-9x=-27$

3

$-90=6y$

$-72=12y$

−7

$-16p=-64$

$-8m=-56$

7

$0.25z=3.25$

$0.75a=11.25$

15

$-3x=0$

$4x=0$

0

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

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

$\frac{z}{2}=14$

28

$-20=\frac{\phantom{\rule{0.4em}{0ex}}q}{-5}$

$\frac{\phantom{\rule{0.4em}{0ex}}c}{-3}=-12$

36

$\frac{y}{9}=-6$

$\frac{q}{6}=-8$

−48

$\frac{m}{-12}=5$

$-4=\frac{p}{-20}$

80

$\frac{2}{3}\phantom{\rule{0.1em}{0ex}}y=18$

$\frac{3}{5}\phantom{\rule{0.1em}{0ex}}r=15$

25

$-\frac{5}{8}\phantom{\rule{0.1em}{0ex}}w=40$

$24=-\frac{3}{4}\phantom{\rule{0.1em}{0ex}}x$

−32

$-\frac{2}{5}=\frac{1}{10}\phantom{\rule{0.1em}{0ex}}a$

$-\frac{1}{3}\phantom{\rule{0.1em}{0ex}}q=-\frac{5}{6}$

5/2

Solve Equations That Need to be Simplified

In the following exercises, solve the equation.

$8a+3a-6a=-17+27$

$6y-3y+12y=-43+28$

y = −1

$-9x-9x+2x=50-2$

$-5m+7m-8m=-6+36$

m = −5

$100-16=4p-10p-p$

$-18-7=5t-9t-6t$

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

$\frac{7}{8}\phantom{\rule{0.1em}{0ex}}n-\frac{3}{4}\phantom{\rule{0.1em}{0ex}}n=9+2$

$\frac{5}{12}\phantom{\rule{0.1em}{0ex}}q+\frac{1}{2}\phantom{\rule{0.1em}{0ex}}q=25-3$

q = 24

$0.25d+0.10d=6-0.75$

$0.05p-0.01p=2+0.24$

p = 56

## 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 $3b=18.$

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 $4g=24.$

6 children

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

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

$1.08 Fuel economy Tania’s SUV gets half as many miles per gallon (mpg) as her husband’s hybrid car. The SUV gets $\text{18 mpg}.$ Find the miles per gallons, $m,$ of the hybrid car, by solving the equation $\frac{1}{2}\phantom{\rule{0.1em}{0ex}}m=18.$ 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 $\frac{1}{3}\phantom{\rule{0.1em}{0ex}}f=14.$ 42 yards ## Writing exercises Frida started to solve the equation $-3x=36$ by adding $3$ to both sides. Explain why Frida’s method will result in the correct solution. Emiliano thinks $x=40$ is the solution to the equation $\frac{1}{2}\phantom{\rule{0.1em}{0ex}}x=80.$ Explain why he is wrong. Answer will vary. ## Self check After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. After reviewing this checklist, what will you do to become confident for all objectives? #### Questions & Answers 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 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 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 what is the Synthesis, properties,and applications of carbon nano chemistry Abhijith Reply 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? s. Reply 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 ? SUYASH Reply for screen printed electrodes ? SUYASH What is lattice structure? s. Reply 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 Sanket Reply what's the easiest and fastest way to the synthesize AgNP? Damian Reply China Cied how did you get the value of 2000N.What calculations are needed to arrive at it Smarajit Reply Privacy Information Security Software Version 1.1a Good Berger describes sociologists as concerned with Mueller Reply 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. Kimberly Reply 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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