# 2.3 Solving equations using the subtraction and addition properties  (Page 3/6)

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Think about twin brothers Andy and Bobby. They are $17$ years old. How old was Andy $3$ years ago? He was $3$ years less than $17,$ so his age was $17-3,$ or $14.$ What about Bobby’s age $3$ years ago? Of course, he was $14$ also. Their ages are equal now, and subtracting the same quantity from both of them resulted in equal ages $3$ years ago.

$\begin{array}{c}a=b\\ a-3=b-3\end{array}$

## Solve an equation using the subtraction property of equality.

1. Use the Subtraction Property of Equality to isolate the variable.
2. Simplify the expressions on both sides of the equation.
3. Check the solution.

Solve: $x+8=17.$

## Solution

We will use the Subtraction Property of Equality to isolate $x.$

 Subtract 8 from both sides. Simplify.

Since $x=9$ makes $x+8=17$ a true statement, we know $9$ is the solution to the equation.

Solve:

$x+6=19$

x = 13

Solve:

$x+9=14$

x = 5

Solve: $100=y+74.$

## Solution

To solve an equation, we must always isolate the variable—it doesn’t matter which side it is on. To isolate $y,$ we will subtract $74$ from both sides.

 Subtract 74 from both sides. Simplify. Substitute $26$ for $y$ to check.

Since $y=26$ makes $100=y+74$ a true statement, we have found the solution to this equation.

Solve:

$95=y+67$

y = 28

Solve:

$91=y+45$

y = 46

## Solve equations using the addition property of equality

In all the equations we have solved so far, a number was added to the variable on one side of the equation. We used subtraction to “undo” the addition in order to isolate the variable.

But suppose we have an equation with a number subtracted from the variable, such as $x-5=8.$ We want to isolate the variable, so to “undo” the subtraction we will add the number to both sides.

We use the Addition Property of Equality, which says we can add the same number to both sides of the equation without changing the equality. Notice how it mirrors the Subtraction Property of Equality.

## Addition property of equality

For any numbers $a,b,$ and $c,$ if

$a=b$

then

$a+c=b+c$

Remember the $17\text{-year-old}$ twins, Andy and Bobby? In ten years, Andy’s age will still equal Bobby’s age. They will both be $27.$

$\begin{array}{c}a=b\\ a+10=b+10\end{array}$

We can add the same number to both sides and still keep the equality.

## Solve an equation using the addition property of equality.

1. Use the Addition Property of Equality to isolate the variable.
2. Simplify the expressions on both sides of the equation.
3. Check the solution.

Solve: $x-5=8.$

## Solution

We will use the Addition Property of Equality to isolate the variable.

 Add 5 to both sides. Simplify.

Solve:

$x-9=13$

x = 22

Solve:

$y-1=3$

y = 4

Solve: $27=a-16.$

## Solution

We will add $16$ to each side to isolate the variable.

 Add 16 to each side. Simplify.

The solution to $27=a-16$ is $a=43.$

Solve:

$19=a-18$

a = 37

Solve:

$27=n-14$

n = 41

## Translate word phrases to algebraic equations

Remember, an equation has an equal sign between two algebraic expressions. So if we have a sentence that tells us that two phrases are equal, we can translate it into an equation. We look for clue words that mean equals . Some words that translate to the equal sign are:

• is equal to
• is the same as
• is
• gives
• was
• will be

It may be helpful to put a box around the equals word(s) in the sentence to help you focus separately on each phrase. Then translate each phrase into an expression, and write them on each side of the equal sign.

#### Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
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
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
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what is nanomaterials​ and their applications of sensors.
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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)=
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