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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses solving equations of the form a x = b size 12{"ax"=b} {} and x a = b size 12{ { {x} over {a} } =b} {} . By the end of the module students should be familiar with the multiplication/division property of equality, be able to solve equations of the form ax = b size 12{ ital "ax"=b} {} and x a = b size 12{ { {x} over {a} } =b} {} and be able to use combined techniques to solve equations.

Section overview

  • Multiplication/ Division Property of Equality
  • Combining Techniques in Equations Solving

Multiplication/ division property of equality

Recall that the equal sign of an equation indicates that the number represented by the expression on the left side is the same as the number represented by the expression on the right side. From this, we can suggest the multiplication/division property of equality.

Multiplication/division property of equality

Given any equation,

  1. We can obtain an equivalent equation by multiplying both sides of the equa­tion by the same nonzero number, that is, if c 0 size 12{c<>0} {} , then a = b size 12{a=b} {} is equivalent to
    a c = b c size 12{a cdot c=b cdot c} {}
  2. We can obtain an equivalent equation by dividing both sides of the equation by the same nonzero number , that is, if c 0 size 12{c<>0} {} , then a = b size 12{a=b} {} is equivalent to
    a c = b c size 12{ { {a} over {c} } = { {b} over {c} } } {}

The multiplication/division property of equality can be used to undo an association with a number that multiplies or divides the variable.

Sample set a

Use the multiplication / division property of equality to solve each equation.

6 y = 54 size 12{6y="54"} {}
6 is associated with y by multiplication. Undo the association by dividing both sides by 6

6 y 6 = 54 6 6 y 6 = 54 9 6 y = 9 alignl { stack { size 12{ { {6y} over {6} } = { {"54"} over {6} } } {} #size 12{ { { { {6}}y} over { { {6}}} } = { { { { {5}} { {4}}} cSup { size 8{9} } } over {6} } } {} # {} #y=9 {} } } {}

Check: When y = 9 size 12{y=9} {}

6 y = 54 size 12{6y="54"} {}

becomes
Does 6 times 9 equal 54? Yes. ,
a true statement.

The solution to 6 y = 54 size 12{6y="54"} {} is y = 9 size 12{y=9} {} .

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x 2 = 27 size 12{ { {x} over {-2} } ="27"} {} .
-2 is associated with x size 12{x} {} by division. Undo the association by multiplying both sides by -2.

2 x 2 = 2 27 alignl { stack { size 12{ left (-2 right ) { {x} over {-2} } = left (-2 right )"27"} {} #{} } } {}

-2 x -2 = 2 27 alignl { stack { size 12{ left ( - 2 right ) { {x} over { - 2} } = left ( - 2 right )"27"} {} #{} } } {}

x = 54 size 12{x= - "54"} {}

Check: When x = 54 size 12{x= - "54"} {} ,

x 2 = 27 size 12{ { {x} over { - 2} } ="27"} {}

becomes
Does negative 54 over negative 2 equal 27? Yes.
a true statement.

The solution to x 2 = 27 size 12{ { {2} over { - 2} } ="27"} {} is x = 54 size 12{x= - "54"} {}

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3 a 7 = 6 size 12{ { {3a} over {7} } =6} {} .
We will examine two methods for solving equations such as this one.

Method 1: Use of dividing out common factors.

3 a 7 = 6 size 12{ { {3a} over {7} } =6} {}
7 is associated with a size 12{a} {} by division. Undo the association by multiplying both sides by 7.

7 3 a 7 = 7 6 size 12{7 cdot { {3a} over {7} } =7 cdot 6} {}
Divide out the 7’s.

7 3 a 7 = 42 size 12{ { {7}} cdot { {3a} over { { {7}}} } ="42"} {}

3 a = 42 size 12{3a="42"} {}
3 is associated with a size 12{a} {} by multiplication. Undo the association by dviding both sides by 3.

3 a 3 = 42 3 size 12{ { {3a} over {3} } = { {"42"} over {3} } } {}

3 a 3 = 14 size 12{ { { { {3}}a} over { { {3}}} } ="14"} {}

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

Check: When a = 14 size 12{a="14"} {} ,

3 a 7 = 6 size 12{ { {3a} over {7} } =6} {}

becomes
Does the quantity 3 times 14, divided by 7 equal 6? Yes. ,
a true statement.

The solution to 3 a 7 = 6 size 12{ { {3a} over {7} } =6} {} is a = 14 size 12{a="14"} {} .

Method 2: Use of reciprocals

Recall that if the product of two numbers is 1, the numbers are reciprocals . Thus 3 7 size 12{ { {3} over {7} } } {} and 7 3 size 12{ { {7} over {3} } } {} are reciprocals.

3 a 7 = 6 size 12{ { {3a} over {7} } =6} {}
Multiply both sides of the equation by 7 3 size 12{ { {7} over {3} } } {} , the reciprocal of 3 7 size 12{ { {3} over {7} } } {} .

7 3 3 a 7 = 7 3 6 size 12{ { {7} over {3} } cdot { {3a} over {7} } = { {7} over {3} } cdot 6} {}

7 1 3 1 3 a 1 7 1 = 7 3 1 6 2 1 size 12{ { { { { {7}}} cSup { size 8{1} } } over { { { {3}}} cSub { size 8{1} } } } cdot { { { { {3}}a} cSup { size 8{1} } } over { { { {7}}} cSub { size 8{1} } } } = { {7} over { { { {3}}} cSub { size 8{1} } } } cdot { { { { {6}}} cSup { size 8{2} } } over {1} } } {}

1 a = 14 a = 14 alignl { stack { size 12{1 cdot a="14"} {} #size 12{a="14"} {} } } {}

Notice that we get the same solution using either method.

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8 x = 24 size 12{-8x="24"} {}
-8 is associated with x by multiplication. Undo the association by dividing both sides by -8.

8 x 8 = 24 8 alignl { stack { size 12{ { {-8x} over {-8} } = { {"24"} over {-8} } } {} #{} } } {}

8 x 8 = 24 8 size 12{ { {-8x} over {-8} } = { {"24"} over {-8} } } {}

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

Check: When x = 3 size 12{x= - 3} {} ,

8 x = 24 size 12{ - 8x="24"} {}

becomes
Does negative 8 times negative 3 equal 24? Yes. ,
a true statement.

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x = 7 . size 12{-x=7 "." } {}
Since x is actually 1 x size 12{-1 cdot x} {} and 1 1 = 1 size 12{ left (-1 right ) left (-1 right )=1} {} , we can isolate x by multiplying both sides of the equation by 1 size 12{-1} {} .

1 x = - 1 7 x = - 7 alignl { stack { size 12{ left (-1 right ) left (-x right )"=-"1 cdot 7} {} #size 12{x"=-"7} {} } } {}

Check: When x = 7 size 12{x=7} {} ,

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

becomes

The solution to x = 7 size 12{ - x=7} {} is x = 7 size 12{x= - 7} {} .

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Practice set a

Use the multiplication/division property of equality to solve each equation. Be sure to check each solution.

7 x = 21 size 12{7x="21"} {}

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

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5 x = 65 size 12{-5x="65"} {}

x = - 13 size 12{x"=-""13"} {}

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

x = - 32 size 12{x"=-""32"} {}

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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
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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
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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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