# 8.2 Solve equations using the division and multiplication properties

 Page 1 / 2
By the end of this section, you will be able to:
• Solve equations using the Division and Multiplication Properties of Equality
• Solve equations that need to be simplified

Before you get started, take this readiness quiz.

1. Simplify: $-7\left(\frac{1}{-7}\right).$
If you missed this problem, review Multiply and Divide Fractions .
2. What is the reciprocal of $-\frac{3}{8}?$
If you missed this problem, review Multiply and Divide Fractions .
3. Evaluate $9x+2$ when $x=-3.$
If you missed this problem, review Multiply and Divide Integers .

## Solve equations using the division and multiplication properties of equality

We introduced the Multiplication and Division Properties of Equality in Solve Equations Using Integers; The Division Property of Equality and Solve Equations with Fractions . We modeled how these properties worked using envelopes and counters and then applied them to solving equations (See Solve Equations Using Integers; The Division Property of Equality ). We restate them again here as we prepare to use these properties again.

## 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 $\frac{a}{c}=\frac{b}{c}.$

Multiplication Property of Equality : For all real numbers $a,b,c,$ if $a=b,$ then $ac=bc.$

When you divide or multiply both sides of an equation by the same quantity, you still have equality.

Let’s review how these properties of equality can be applied in order to solve equations. Remember, the goal is to ‘undo’ the operation on the variable. In the example below the variable is multiplied by $4,$ so we will divide both sides by $4$ to ‘undo’ the multiplication.

Solve: $4x=-28.$

## Solution

We use the Division Property of Equality to divide both sides by $4.$

 Divide both sides by 4 to undo the multiplication. Simplify. Check your answer. Let $x=-7$ .

Since this is a true statement, $x=-7$ is a solution to $4x=-28.$

Solve: $3y=-48.$

y = −16

Solve: $4z=-52.$

z = −13

In the previous example, to ‘undo’ multiplication, we divided. How do you think we ‘undo’ division?

Solve: $\frac{\phantom{\rule{0.4em}{0ex}}a}{-7}=-42.$

## Solution

Here $a$ is divided by $-7.$ We can multiply both sides by $-7$ to isolate $a.$

 Multiply both sides by $-7$ . Simplify. Check your answer. Let $a=294$ .

Solve: $\frac{\phantom{\rule{0.4em}{0ex}}b}{-6}=-24.$

b = 144

Solve: $\frac{\phantom{\rule{0.4em}{0ex}}c}{-8}=-16.$

c = 128

Solve: $-r=2.$

## Solution

Remember $-r$ is equivalent to $-1r.$

 Rewrite $-r$ as $-1r$ . Divide both sides by $-1$ . Check. Substitute $r=-2$ Simplify.

In Solve Equations with Fractions , we saw that there are two other ways to solve $-r=2.$

We could multiply both sides by $-1.$

We could take the opposite of both sides.

Solve: $-k=8.$

k = −8

Solve: $-g=3.$

g = −3

Solve: $\frac{2}{3}\phantom{\rule{0.1em}{0ex}}x=18.$

## Solution

Since the product of a number and its reciprocal is $1,$ our strategy will be to isolate $x$ by multiplying by the reciprocal of $\frac{2}{3}.$

 Multiply by the reciprocal of $\frac{2}{3}$ . Reciprocals multiply to one. Multiply. Check your answer. Let $x=27$

Notice that we could have divided both sides of the equation $\frac{2}{3}\phantom{\rule{0.1em}{0ex}}x=18$ by $\frac{2}{3}$ to isolate $x.$ While this would work, multiplying by the reciprocal requires fewer steps.

Solve: $\frac{2}{5}\phantom{\rule{0.1em}{0ex}}n=14.$

n = 35

Solve: $\frac{5}{6}\phantom{\rule{0.1em}{0ex}}y=15.$

y = 18

## Solve equations that need to be simplified

Many equations start out more complicated than the ones we’ve just solved. First, we need to simplify both sides of the equation as much as possible

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
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
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
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
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
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?
what is a peer
What is meant by 'nano scale'?
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 ?
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?
what king of growth are you checking .?
Renato
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
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.
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.
What is the expressiin for seven less than four times the number of nickels
How do i figure this problem out.
how do you translate this in Algebraic Expressions
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
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?