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

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
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?
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
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
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?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
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?