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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

if theta =30degree so COS2 theta = 1- 10 square theta upon 1 + tan squared theta
Martin Reply
how to compute this 1. g(1-x) 2. f(x-2) 3. g (-x-/5) 4. f (x)- g (x)
Yanah Reply
hi
John
hi
Grace
what sup friend
John
not much For functions, there are two conditions for a function to be the inverse function:   1--- g(f(x)) = x for all x in the domain of f     2---f(g(x)) = x for all x in the domain of g Notice in both cases you will get back to the  element that you started with, namely, x.
Grace
sin theta=3/4.prove that sec square theta barabar 1 + tan square theta by cosec square theta minus cos square theta
Umesh Reply
acha se dhek ke bata sin theta ke value
Ajay
sin theta ke ja gha sin square theta hoga
Ajay
I want to know trigonometry but I can't understand it anyone who can help
Siyabonga Reply
Yh
Idowu
which part of trig?
Nyemba
functions
Siyabonga
trigonometry
Ganapathi
differentiation doubhts
Ganapathi
hi
Ganapathi
hello
Brittany
Prove that 4sin50-3tan 50=1
Sudip Reply
f(x)= 1 x    f(x)=1x  is shifted down 4 units and to the right 3 units.
Sebit Reply
f (x) = −3x + 5 and g (x) = x − 5 /−3
Sebit
what are real numbers
Marty Reply
I want to know partial fraction Decomposition.
Adama Reply
classes of function in mathematics
Yazidu Reply
divide y2_8y2+5y2/y2
Sumanth Reply
wish i knew calculus to understand what's going on 🙂
Dashawn Reply
@dashawn ... in simple terms, a derivative is the tangent line of the function. which gives the rate of change at that instant. to calculate. given f(x)==ax^n. then f'(x)=n*ax^n-1 . hope that help.
Christopher
thanks bro
Dashawn
maybe when i start calculus in a few months i won't be that lost 😎
Dashawn
what's the derivative of 4x^6
Axmed Reply
24x^5
James
10x
Axmed
24X^5
Taieb
Thanks for this helpfull app
Axmed Reply
secA+tanA=2√5,sinA=?
richa Reply
tan2a+tan2a=√3
Rahulkumar
classes of function
Yazidu
if sinx°=sin@, then @ is - ?
NAVJIT Reply
the value of tan15°•tan20°•tan70°•tan75° -
NAVJIT
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
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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