<< Chapter < Page Chapter >> Page >

Addition/subtraction property of equality

From this, we can suggest the addition/subtraction property of equality .
Given any equation,
  1. We can obtain an equivalent equation by adding the same number to both sides of the equation.
  2. We can obtain an equivalent equation by subtracting the same number from both sides of the equation.

The idea behind equation solving

The idea behind equation solving is to isolate the variable on one side of the equation. Signs of operation (+, -, ⋅,÷) are used to associate two numbers. For example, in the expression 5 + 3 size 12{5+3} {} , the numbers 5 and 3 are associated by addition. An association can be undone by performing the opposite operation. The addition/subtraction property of equality can be used to undo an association that is made by addition or subtraction.

Subtraction is used to undo an addition.

Addition is used to undo a subtraction.

The procedure is illustrated in the problems of [link] .

Sample set b

Use the addition/subtraction property of equality to solve each equation.

x + 4 = 6 size 12{x+4=6} {} .
4 is associated with x size 12{x} {} by addition. Undo the association by subtracting 4 from both sides.

x + 4 4 = 6 4 x + 0 = 2 x = 2 alignl { stack { size 12{x+4 - 4=6 - 4} {} #size 12{x+0=2} {} # size 12{x=2} {}} } {}

Check: When x = 2 size 12{x=2} {} , x + 4 size 12{x+4} {} becomes

Does 2 + 4 = 6? Yes.

The solution to x + 4 = 6 size 12{x+4=6} {} is x = 2 size 12{x=2} {} .

Got questions? Get instant answers now!

m 8 = 5 size 12{m - 8=5} {} . 8 is associated with m size 12{m} {} by subtraction. Undo the association by adding 8 to both sides.

m 8 + 8 = 5 + 8 m + 0 = 13 m = 13 alignl { stack { size 12{m - 8+8=5+8} {} #size 12{m+0="13"} {} # size 12{m="13"} {}} } {}

Check: When m = 13 size 12{m="13"} {} ,

becomes
m - 8 = 5. After substituting, does 13 - 8 = 5? Yes.
a true statement.

The solution to m 8 = 5 size 12{m - 8=5} {} is m = 13 size 12{m="13"} {} .

Got questions? Get instant answers now!

3 5 = y 2 + 8 size 12{ - 3 - 5=y - 2+8} {} . Before we use the addition/subtraction property, we should simplify as much as possible.

3 5 = y 2 + 8 size 12{ - 3 - 5=y - 2+8} {}

8 = y + 6 size 12{ - 8=y+6} {}
6 is associated with y size 12{y} {} by addition. Undo the association by subtracting 6 from both sides.

8 6 = y + 6 6 14 = y + 0 14 = y alignl { stack { size 12{ - 8 - 6=y+6 - 6} {} #size 12{ - "14"=y+0} {} # size 12{ - "14"=y} {}} } {}
This is equivalent to y = 14 size 12{y= - "14"} {} .

Check: When y = 14 size 12{y= - "14"} {} ,

3 5 = y 2 + 8 size 12{ - 3 - 5=y - 2+8} {}

becomes
Does negative 3 minus 5 equal negative 14 minus 2 plus 8? Yes. ,
a true statement.

The solution to 3 5 = y 2 + 8 size 12{ - 3 - 5=y - 2+8} {} is y = 14 size 12{y= - "14"} {} .

Got questions? Get instant answers now!

5a + 1 + 6a = 2 size 12{ - 5a+1+6a= - 2} {} . Begin by simplifying the left side of the equation.

- 5 a + 1 + 6 a - 5 + 6 = 1 = - 2

a + 1 = 2 size 12{a+1= - 2} {} 1 is associated with a size 12{a} {} by addition. Undo the association by subtracting 1 from both sides.

a + 1 1 = 2 1 a + 0 = 3 a = 3 alignl { stack { size 12{a+1 - 1= - 2 - 1} {} #size 12{a+0= - 3} {} # size 12{a= - 3} {}} } {}

Check: When a = 3 size 12{a= - 3} {} ,

5a + 1 + 6a = 2 size 12{ - 5a+1+6a= - 2} {}

becomes
Does negative 5 times negative 3 plus 1 plus 6 times negative 3 equal negative 2? Yes. ,
a true statement.

The solution to 5 a + 1 + 6 a = 2 size 12{ - 5a+1+6a= - 2} {} is a = 3 size 12{a= - 3} {} .

Got questions? Get instant answers now!

7 k 4 = 6 k + 1 size 12{7k - 4=6k+1} {} . In this equation, the variable appears on both sides. We need to isolate it on one side. Although we can choose either side, it will be more convenient to choose the side with the larger coefficient. Since 8 is greater than 6, we’ll isolate k size 12{k} {} on the left side.

7 k 4 = 6 k + 1 size 12{7k - 4=6k+1} {} Since 6 k size 12{6k} {} represents + 6 k size 12{+6k} {} , subtract 6 k size 12{6k} {} from each side.

7 k - 4 - 6 k 7 - 6 = 1 = 6 k + 1 - 6 k 6 - 6 = 0

k 4 = 1 size 12{k - 4=1} {} 4 is associated with k size 12{k} {} by subtraction. Undo the association by adding 4 to both sides.

k 4 + 4 = 1 + 4 k = 5 alignl { stack { size 12{k - 4+4=1+4} {} #size 12{k=5} {} } } {}

Check: When k = 5 size 12{k=5} {} ,

7 k 4 = 6 k + 1 size 12{7k - 4=6k+1} {}

becomes
Does 7 times 5 minus 4 equal 6 times 5 plus 1? Yes.
a true statement.

The solution to 7 k 4 = 6 k + 1 size 12{7k - 4=6k+1} {} is k = 5 size 12{k=5} {} .

Got questions? Get instant answers now!

8 + x = 5 size 12{ - 8+x=5} {} . -8 is associated with x size 12{x} {} by addition. Undo the by subtracting -8 from both sides. Subtracting -8 we get 8 =+ 8 size 12{ - left ( - 8 right )"=+"8} {} . We actually add 8 to both sides.

8 + x + 8 = 5 + 8 size 12{ - 8+x+8=5+8} {}

x = 13 size 12{x="13"} {}

Check: When x = 13 size 12{x="13"} {}

8 + x = 5 size 12{ - 8+x=5} {}

becomes
Does negative 8 plus 13 equal 5? Yes. ,
a true statement.

The solution to 8 + x = 5 size 12{ - 8+x=5} {} is x = 13 size 12{x="13"} {} .

Got questions? Get instant answers now!

Practice set b

y + 9 = 4 size 12{y+9=4} {}

y = 5 size 12{y= - 5} {}

Got questions? Get instant answers now!

a 4 = 11 size 12{a - 4="11"} {}

a = 15 size 12{a="15"} {}

Got questions? Get instant answers now!

1 + 7 = x + 3 size 12{ - 1+7=x+3} {}

x = 3 size 12{x=3} {}

Got questions? Get instant answers now!

8 m + 4 7 m = 2 3 size 12{8m+4 - 7m= left ( - 2 right ) left ( - 3 right )} {}

m = 2 size 12{m=2} {}

Got questions? Get instant answers now!

12 k 4 = 9 k 6 + 2 k size 12{"12"k - 4=9k - 6+2k} {}

k = 2 size 12{k= - 2} {}

Got questions? Get instant answers now!

3 + a = 4 size 12{ - 3+a= - 4} {}

a = 1 size 12{a= - 1} {}

Got questions? Get instant answers now!

Exercises

For the following 10 problems, verify that each given value is a solution to the given equation.

x 11 = 5 size 12{x - "11"=5} {} , x = 16 size 12{x="16"} {}

Substitute x = 4 size 12{x=4} {} into the equation 4 x 11 = 5 size 12{4x - "11"=5} {} .
16 11 = 5 5 = 5 alignl { stack { size 12{"16" - "11"=5} {} #size 12{5=5} {} } } {}
x = 4 size 12{x=4} {} is a solution.

Got questions? Get instant answers now!

y 4 = 6 size 12{y - 4= - 6} {} , y = 2 size 12{y= - 2} {}

Got questions? Get instant answers now!

2 m 1 = 1 size 12{2m - 1=1} {} , m = 1 size 12{m=1} {}

Substitute m = 1 size 12{m=1} {} into the equation 2 m 1 = 1 size 12{2m - 1=1} {} .
Does 2 minus 1 equal 1? Yes.
m = 1 size 12{m=1} {} is a solution.

Got questions? Get instant answers now!

5 y + 6 = 14 size 12{5y+6= - "14"} {} , y = 4 size 12{y= - 4} {}

Got questions? Get instant answers now!

3x + 2 7x = 5x 6 size 12{3x+2 - 7x= - 5x - 6} {} , x = 8 size 12{x= - 8} {}

Substitute x = 8 size 12{x= - 8} {} into the equation 3x + 2 7 = 5x 6 size 12{3x+2 - 7= - 5x - 6} {} .
Does negative 24 plus 2 minus 7 equal 40 minus 6? Yes.
x = 8 size 12{x= - 8} {} is a solution.

Got questions? Get instant answers now!

6 a + 3 + 3 a = 4 a + 7 3 a size 12{ - 6a+3+3a=4a+7 - 3a} {} , a = 1 size 12{a= - 1} {}

Got questions? Get instant answers now!

8 + x = 8 size 12{ - 8+x= - 8} {} , x = 0 size 12{x=0} {}

Substitute x = 0 size 12{x=0} {} into the equation 8 + x = 8 size 12{ - 8+x= - 8} {} .
Does negative 8 plus 0 equal negative 8? Yes.
x = 0 size 12{x=0} {} is a solution.

Got questions? Get instant answers now!

8 b + 6 = 6 5 b size 12{8b+6=6 - 5b} {} , b = 0 size 12{b=0} {}

Got questions? Get instant answers now!

4 x 5 = 6 x 20 size 12{4x - 5=6x - "20"} {} , x = 15 2 size 12{x= { {"15"} over {2} } } {}

Substitute x = 15 2 size 12{x= { {"15"} over {2} } } {} into the equation 4 x 5 = 6 x 20 size 12{4x - 5=6x - "20"} {} .
Does 30 minus 5 equal 45 minus 20? Yes.
x = 15 2 size 12{x= { {"15"} over {2} } } {} is a solution.

Got questions? Get instant answers now!

3 y + 7 = 2 y 15 size 12{ - 3y+7=2y - "15"} {} , y = 22 5 size 12{y= { {"22"} over {5} } } {}

Got questions? Get instant answers now!

Solve each equation. Be sure to check each result.

y 6 = 5 size 12{y - 6=5} {}

y = 11 size 12{y="11"} {}

Got questions? Get instant answers now!

m + 8 = 4 size 12{m+8=4} {}

Got questions? Get instant answers now!

k 1 = 4 size 12{k - 1=4} {}

k = 5 size 12{k=5} {}

Got questions? Get instant answers now!

h 9 = 1 size 12{h - 9=1} {}

Got questions? Get instant answers now!

a + 5 = 4 size 12{a+5= - 4} {}

a = 9 size 12{a= - 9} {}

Got questions? Get instant answers now!

b 7 = 1 size 12{b - 7= - 1} {}

Got questions? Get instant answers now!

x + 4 9 = 6 size 12{x+4 - 9=6} {}

x = 11 size 12{x="11"} {}

Got questions? Get instant answers now!

y 8 + 10 = 2 size 12{y - 8+"10"=2} {}

Got questions? Get instant answers now!

z + 6 = 6 size 12{z+6=6} {}

z = 0 size 12{z=0} {}

Got questions? Get instant answers now!

w 4 = 4 size 12{w - 4= - 4} {}

Got questions? Get instant answers now!

x + 7 9 = 6 size 12{x+7 - 9=6} {}

x = 8 size 12{x=8} {}

Got questions? Get instant answers now!

y 2 + 5 = 4 size 12{y - 2+5=4} {}

Got questions? Get instant answers now!

m + 3 8 = 6 + 2 size 12{m+3 - 8= - 6+2} {}

m = 1 size 12{m=1} {}

Got questions? Get instant answers now!

z + 10 8 = 8 + 10 size 12{z+"10" - 8= - 8+"10"} {}

Got questions? Get instant answers now!

2 + 9 = k 8 size 12{2+9=k - 8} {}

k = 19 size 12{k="19"} {}

Got questions? Get instant answers now!

5 + 3 = h 4 size 12{ - 5+3=h - 4} {}

Got questions? Get instant answers now!

3 m 4 = 2 m + 6 size 12{3m - 4=2m+6} {}

m = 10 size 12{m="10"} {}

Got questions? Get instant answers now!

5 a + 6 = 4 a 8 size 12{5a+6=4a - 8} {}

Got questions? Get instant answers now!

8 b + 6 + 2 b = 3 b 7 + 6 b 8 size 12{8b+6+2b=3b - 7+6b - 8} {}

b = 21 size 12{b= - "21"} {}

Got questions? Get instant answers now!

12 h 1 3 5 h = 2 h + 5 h + 3 ( 4 ) size 12{"12"h - 1 - 3 - 5h=2h+5h+3 \( - 4 \) } {}

Got questions? Get instant answers now!

4 a + 5 2 a = 3 a 11 2 a size 12{ - 4a+5 - 2a= - 3a - "11" - 2a} {}

a = 16 size 12{a="16"} {}

Got questions? Get instant answers now!

9 n 2 6 + 5 n = 3 n 2 5 6 n size 12{ - 9n - 2 - 6+5n=3n - left (2 right ) left ( - 5 right ) - 6n} {}

Got questions? Get instant answers now!

Calculator exercises

y 2 . 161 = 5 . 063 size 12{y - 2 "." "161"=5 "." "063"} {}

y = 7 . 224 size 12{y=7 "." "224"} {}

Got questions? Get instant answers now!

a 44 . 0014 = 21 . 1625 size 12{a - "44" "." "0014"= - "21" "." "1625"} {}

Got questions? Get instant answers now!

0 . 362 0 . 416 = 5 . 63 m 4 . 63 m size 12{ - 0 "." "362" - 0 "." "416"=5 "." "63"m - 4 "." "63"m} {}

m = 0 . 778 size 12{m= - 0 "." "778"} {}

Got questions? Get instant answers now!

8 . 078 9 . 112 = 2 . 106 y 1 . 106 y size 12{8 "." "078" - 9 "." "112"=2 "." "106"y - 1 "." "106"y} {}

Got questions? Get instant answers now!

4 . 23 k + 3 . 18 = 3 . 23 k 5 . 83 size 12{4 "." "23"k+3 "." "18"=3 "." "23"k - 5 "." "83"} {}

k = 9 . 01 size 12{k= - 9 "." "01"} {}

Got questions? Get instant answers now!

6 . 1185 x 4 . 0031 = 5 . 1185 x 0 . 0058 size 12{6 "." "1185"x - 4 "." "0031"=5 "." "1185"x - 0 "." "0058"} {}

Got questions? Get instant answers now!

21 . 63 y + 12 . 40 5 . 09 y = 6 . 11 y 15 . 66 + 9 . 43 y size 12{"21" "." "63"y+"12" "." "40" - 5 "." "09"y=6 "." "11"y - "15" "." "66"+9 "." "43"y} {}

y = 28 . 06 size 12{y= - "28" "." "06"} {}

Got questions? Get instant answers now!

0 . 029 a 0 . 013 0 . 034 0 . 057 = 0 . 038 + 0 . 56 + 1 . 01 a size 12{0 "." "029"a - 0 "." "013" - 0 "." "034" - 0 "." "057"= - 0 "." "038"+0 "." "56"+1 "." "01"a} {}

Got questions? Get instant answers now!

Exercises for review

( [link] ) Is 7 calculators 12 students size 12{ { {7" calculators"} over {"12"" students"} } } {} an example of a ratio or a rate?

rate

Got questions? Get instant answers now!

( [link] ) Convert 3 8 size 12{ {3} over {8} } {} % to a decimal.

Got questions? Get instant answers now!

( [link] ) 0.4% of what number is 0.014?

3.5

Got questions? Get instant answers now!

( [link] ) Use the clustering method to estimate the sum: 89 + 93 + 206 + 198 + 91 size 12{"89"+"93"+"206"+"198"+"91"} {}

Got questions? Get instant answers now!

( [link] ) Combine like terms: 4 x + 8 y + 12 y + 9 x 2 y size 12{4x+8y+"12"y+9x - 2y} {} .

13 x + 18 y size 12{"13"x+"18"y} {}

Got questions? Get instant answers now!

Questions & Answers

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
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
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
From 1973 to 1979, in the United States, there was an increase of 166.6% of Ph.D. social scien­tists to 52,000. How many were there in 1973?
Khizar Reply
7hours 36 min - 4hours 50 min
Tanis Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Fundamentals of mathematics' conversation and receive update notifications?

Ask