# 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 Application of nanotechnology in medicine what is variations in raman spectra for nanomaterials Jyoti Reply I only see partial conversation and what's the question here! Crow Reply what about nanotechnology for water purification RAW Reply please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment. Damian yes that's correct Professor I think Professor 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 if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION Anam analytical skills graphene is prepared to kill any type viruses . Anam 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 hi Loga what does nano mean? Anassong Reply nano basically means 10^(-9). nanometer is a unit to measure length. Bharti 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 what is hormones? Wellington 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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