# 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 how can chip be made from sand Eke Reply are nano particles real Missy Reply yeah Joseph Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master? Lale Reply no can't Lohitha where we get a research paper on Nano chemistry....? Maira Reply nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review Ali what are the products of Nano chemistry? Maira Reply There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others.. learn Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level learn Google da no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts Bhagvanji hey Giriraj Preparation and Applications of Nanomaterial for Drug Delivery Hafiz Reply revolt da Application of nanotechnology in medicine has a lot of application modern world Kamaluddeen yes narayan what is variations in raman spectra for nanomaterials Jyoti Reply ya I also want to know the raman spectra Bhagvanji 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 Nasa has use it in the 60's, copper as water purification in the moon travel. Alexandre nanocopper obvius Alexandre 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 Any one who tell me about Preparation and application of Nanomaterial for drug Delivery Hafiz 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 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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