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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} {} .

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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"} {} .

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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"} {} .

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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} {} .

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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} {} .

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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"} {} .

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Practice set b

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

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

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

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

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

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

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

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

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

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

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

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

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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.

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

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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.

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

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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.

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

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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.

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

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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.

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

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Solve each equation. Be sure to check each result.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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"} {}

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

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

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

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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} {}

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Calculator exercises

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

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

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

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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"} {}

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

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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"} {}

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

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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"} {}

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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} {}

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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

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

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

3.5

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

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( [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} {}

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Questions & Answers

what is variations in raman spectra for nanomaterials
Jyoti Reply
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
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
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
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
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?
Damian Reply
absolutely yes
Daniel
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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