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3 x 8 = 6 size 12{ { {3x} over {8} } =6} {}

x = 16 size 12{x="16"} {}

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y = 3 size 12{-y=3} {}

y = - 3 size 12{y"=-"3} {}

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

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

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Combining techniques in equation solving

Having examined solving equations using the addition/subtraction and the multi­plication/division principles of equality, we can combine these techniques to solve more complicated equations.

When beginning to solve an equation such as 6 x - 4 = 16 size 12{6 ital "x-"4= - "16"} {} , it is helpful to know which property of equality to use first, addition/subtraction or multiplication/di­vision. Recalling that in equation solving we are trying to isolate the variable (disas­sociate numbers from it), it is helpful to note the following.

To associate numbers and letters, we use the order of operations.

  1. Multiply/divide
  2. Add/subtract

To undo an association between numbers and letters, we use the order of opera­tions in reverse.

  1. Add/subtract
  2. Multiply/divide

Sample set b

Solve each equation. (In these example problems, we will not show the checks.)

6 x 4 = - 16 size 12{6x-4"=-""16"} {}
-4 is associated with x by subtraction. Undo the association by adding 4 to both sides.

6 x 4 + 4 = - 16 + 4 size 12{6x-4+4"=-""16"+4} {}

6 x = - 12 size 12{6x"=-""12"} {}
6 is associated with x by multiplication. Undo the association by dividing both sides by 6

6 x 6 = 12 6 size 12{ { {6x} over {6} } = { {-"12"} over {6} } } {}

x = - 2 size 12{x"=-"2} {}

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8 k + 3 = -45 . size 12{-8k+3"=-""45" "." } {}
3 is associated with k by addition. Undo the association by subtracting 3 from both sides.

8 k + 3 3 = -45 3 size 12{-8k+3-3"=-""45"-3} {}

8 k = - 48 size 12{-8k"=-""48"} {}
-8 is associated with k by multiplication. Undo the association by dividing both sides by -8.

8 k 8 = 48 8 size 12{ { {-8k} over {-8} } = { {-"48"} over {-8} } } {}

k = 6 size 12{k=6} {}

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5 m 6 4 m = 4 m 8 + 3 m . size 12{5m-6-4m=4m-8+3m "." } {} Begin by solving this equation by combining like terms.

m 6 = 7 m 8 size 12{m-6=7m-8} {} Choose a side on which to isolate m . Since 7 is greater than 1, we'll isolate m on the right side.
Subtract m from both sides.

m 6 m = 7 m 8 m size 12{-m-6-m=7m-8-m} {}

6 = 6 m 8 size 12{-6=6m-8} {}
8 is associated with m by subtraction. Undo the association by adding 8 to both sides.

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

2 = 6 m size 12{2=6m} {}
6 is associated with m by multiplication. Undo the association by dividing both sides by 6.

2 6 = 6 m 6 size 12{ { {2} over {6} } = { {6m} over {6} } } {} Reduce.

1 3 = m size 12{ { {1} over {3} } =m} {}

Notice that if we had chosen to isolate m on the left side of the equation rather than the right side, we would have proceeded as follows:

m 6 = 7 m 8 size 12{m-6=7m-8} {}
Subtract 7 m from both sides.

m 6 7 m = 7 m 8 7 m size 12{m-6-7m=7m-8-7m} {}

6 m 6 = -8 size 12{-6m-6"=-"8} {}
Add 6 to both sides,

6 m 6 + 6 = -8 + 6 size 12{-6m-6+6"=-"8+6} {}

6 m = - 2 size 12{-6m"=-"2} {}
Divide both sides by -6.

6 m 6 = 2 6 size 12{ { {-6m} over {-6} } = { {-2} over {-6} } } {}

m = 1 3 size 12{m= { {1} over {3} } } {}

This is the same result as with the previous approach.

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8 x 7 = - 2 size 12{ { {8x} over {7} } "=-"2} {}
7 is associated with x by division. Undo the association by multiplying both sides by 7.

7 8 x 7 = 7 2 size 12{7 cdot { {8x} over {7} } =7 left (-2 right )} {}

7 8 x 7 = - 14 size 12{ { {7}} cdot { {8x} over { { {7}}} } "=-""14"} {}

8 x = - 14 size 12{8x"=-""14"} {}
8 is associated with x by multiplication. Undo the association by dividing both sides by 8.

8 x 8 = 7 4 size 12{ { { { {8}}x} over { { {8}}} } = { {-7} over {4} } } {}

x = 7 4 size 12{x= { {-7} over {4} } } {}

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Practice set b

Solve each equation. Be sure to check each solution.

5 m + 7 = - 13 size 12{5m+7"=-""13"} {}

m = - 4 size 12{m"=-"4} {}

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3 a 6 = 9 size 12{-3a-6=9} {}

a = - 5 size 12{a"=-"5} {}

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2 a + 10 3 a = 9 size 12{2a+"10"-3a=9} {}

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

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11 x 4 13 x = 4 x + 14 size 12{"11"x-4-"13"x=4x+"14"} {}

x = - 3 size 12{x"=-"3} {}

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3 m + 8 = - 5 m + 1 size 12{-3m+8"=-"5m+1} {}

m = - 7 2 size 12{m"=-" { {7} over {2} } } {}

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5 y + 8 y 11 = - 11 size 12{5y+8y-"11""=-""11"} {}

y = 0 size 12{y=0} {}

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Exercises

Solve each equation. Be sure to check each result.

7 x = 42 size 12{7x="42"} {}

x = 6 size 12{x=6} {}

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8 x = 81 size 12{8x="81"} {}

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10 x = 120 size 12{"10"x="120"} {}

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

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11 x = 121 size 12{"11"x="121"} {}

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6 a = 48 size 12{-6a="48"} {}

a = - 8 size 12{a"=-"8} {}

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9 y = 54 size 12{-9y="54"} {}

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3 y = - 42 size 12{-3y"=-""42"} {}

y = 14 size 12{y="14"} {}

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5 a = - 105 size 12{-5a"=-""105"} {}

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2 m = - 62 size 12{2m"=-""62"} {}

m = - 31 size 12{m"=-""31"} {}

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3 m = - 54 size 12{3m"=-""54"} {}

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x 4 = 7 size 12{ { {x} over {4} } =7} {}

x = 28 size 12{x="28"} {}

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y 3 = 11 size 12{ { {y} over {3} } ="11"} {}

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z 6 = - 14 size 12{ { {-z} over {6} } "=-""14"} {}

z = 84 size 12{z="84"} {}

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w 5 = 1 size 12{ { {-w} over {5} } =1} {}

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3 m 1 = - 13 size 12{3m-1"=-""13"} {}

m = - 4 size 12{m"=-"4} {}

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4 x + 7 = - 17 size 12{4x+7"=-""17"} {}

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2 + 9 x = - 7 size 12{2+9x"=-"7} {}

x = - 1 size 12{x"=-"1} {}

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5 11 x = 27 size 12{5-"11"x="27"} {}

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32 = 4 y + 6 size 12{"32"=4y+6} {}

y = 13 2 size 12{y= { {"13"} over {2} } } {}

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5 + 4 = - 8 m + 1 size 12{-5+4"=-"8m+1} {}

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3 k + 6 = 5 k + 10 size 12{3k+6=5k+"10"} {}

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

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4 a + 16 = 6 a + 8 a + 6 size 12{4a+"16"=6a+8a+6} {}

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6 x + 5 + 2 x 1 = 9 x 3 x + 15 size 12{6x+5+2x-1=9x-3x+"15"} {}

x = 11 2 or 5 1 2 size 12{x= { {"11"} over {2} } " or 5" { {1} over {2} } } {}

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9 y 8 + 3 y + 7 = - 7 y + 8 y 5 y + 9 size 12{-9y-8+3y+7"=-"7y+8y-5y+9} {}

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3 a = a + 5 size 12{-3a=a+5} {}

a = - 5 4 size 12{a"=-" { {5} over {4} } } {}

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5 b = - 2 b + 8 b + 1 size 12{5b"=-"2b+8b+1} {}

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3 m + 2 8 m 4 = - 14 m + m 4 size 12{-3m+2-8m-4"=-""14"m+m-4} {}

m = - 1 size 12{m"=-"1} {}

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5 a + 3 = 3 size 12{5a+3=3} {}

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7 x + 3 x = 0 size 12{7x+3x=0} {}

x = 0 size 12{x=0} {}

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7 g + 4 11 g = - 4 g + 1 + g size 12{7g+4-"11"g"=-"4g+1+g} {}

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5 a 7 = 10 size 12{ { {5a} over {7} } ="10"} {}

a = 14 size 12{a="14"} {}

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2 m 9 = 4 size 12{ { {2m} over {9} } =4} {}

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3 x 4 = 9 2 size 12{ { {3x} over {4} } = { {9} over {2} } } {}

x = 6 size 12{x=6} {}

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8 k 3 = 32 size 12{ { {8k} over {3} } ="32"} {}

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3 a 8 3 2 = 0 size 12{ { {3a} over {8} } - { {3} over {2} } =0} {}

a = 4 size 12{a=4} {}

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5 m 6 25 3 = 0 size 12{ { {5m} over {6} } - { {"25"} over {3} } =0} {}

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Exercises for review

( [link] ) Use the distributive property to compute 40 28 size 12{"40" cdot "28"} {} .

40 30 2 = 1200 80 = 1120 size 12{"40" left ("30"-2 right )="1200"-"80"="1120"} {}

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( [link] ) Approximating π by 3.14, find the approximate circumference of the circle.

A circle with radius 8cm.

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( [link] ) Find the area of the parallelogram.

A parallelogram with base 20cm and height 11cm.

220 sq cm

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( [link] ) Find the value of 3 4 15 2 5 size 12{ { {-3 left (4-"15" right )-2} over {-5} } } {} .

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( [link] ) Solve the equation x 14 + 8 = - 2 . size 12{x-"14"+8"=-"2 "." } {}

x = 4 size 12{x=4} {}

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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
7hours 36 min - 4hours 50 min
Tanis 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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