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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

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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.
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what is the actual application of fullerenes nowadays?
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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.
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
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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.
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of graphene you mean?
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or in general
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in general
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Graphene has a hexagonal structure
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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
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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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