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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

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 to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
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?
s. Reply
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
Devang Reply
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
Abhijith Reply
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?
s. Reply
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 ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
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
Sanket Reply
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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