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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 inflation
Bright Reply
a general and ongoing rise in the level of prices in an economy
AI-Robot
What are the factors that affect demand for a commodity
Florence Reply
price
Kenu
differentiate between demand and supply giving examples
Lambiv Reply
differentiated between demand and supply using examples
Lambiv
what is labour ?
Lambiv
how will I do?
Venny Reply
how is the graph works?I don't fully understand
Rezat Reply
information
Eliyee
devaluation
Eliyee
t
WARKISA
hi guys good evening to all
Lambiv
multiple choice question
Aster Reply
appreciation
Eliyee
explain perfect market
Lindiwe Reply
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
Ezea
What is ceteris paribus?
Shukri Reply
other things being equal
AI-Robot
When MP₁ becomes negative, TP start to decline. Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab
Kelo
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)
Kelo
yes,thank you
Shukri
Can I ask you other question?
Shukri
what is monopoly mean?
Habtamu Reply
What is different between quantity demand and demand?
Shukri Reply
Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded
Ezea
ok
Shukri
how do you save a country economic situation when it's falling apart
Lilia Reply
what is the difference between economic growth and development
Fiker Reply
Economic growth as an increase in the production and consumption of goods and services within an economy.but Economic development as a broader concept that encompasses not only economic growth but also social & human well being.
Shukri
production function means
Jabir
What do you think is more important to focus on when considering inequality ?
Abdisa Reply
any question about economics?
Awais Reply
sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏
Asui
it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs
Awais
thank you so much 👍 sir
Asui
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p
Cornelius
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities,
Cornelius
Suppose a consumer consuming two commodities X and Y has The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50. A,Calculate quantities of x and y which maximize utility. B,Calculate value of Lagrange multiplier. C,Calculate quantities of X and Y consumed with a given price. D,alculate optimum level of output .
Feyisa Reply
Answer
Feyisa
c
Jabir
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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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