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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to solve equations of the form x + a = b and x - a = b . By the end of the module students should understand the meaning and function of an equation, understand what is meant by the solution to an equation and be able to solve equations of the form x + a = b size 12{x+a=b} {} and x a = b size 12{x - a=b} {} .

Section overview

  • Equations
  • Solutions and Equivalent Equations
  • Solving Equations

Equations

Equation

An equation is a statement that two algebraic expressions are equal.

The following are examples of equations:

x + 6 This expression = = 10 This expression x - 4 This expression = = - 11 This expression 3 y - 5 This expression = = - 2 + 2 y This expression

Notice that x + 6 size 12{x+6} {} , x 4 size 12{x - 4} {} , and 3 y 5 size 12{3y - 5} {} are not equations. They are expressions. They are not equations because there is no statement that each of these expressions is equal to another expression.

Solutions and equivalent equations

Conditional equations

The truth of some equations is conditional upon the value chosen for the variable. Such equations are called conditional equations . There are two additional types of equations. They are examined in courses in algebra, so we will not consider them now.

Solutions and solving an equation

The set of values that, when substituted for the variables, make the equation true, are called the solutions of the equation.
An equation has been solved when all its solutions have been found.

Sample set a

Verify that 3 is a solution to x + 7 = 10 size 12{x+7 = "10"} {} .

When x = 3 size 12{x=3} {} ,

x + 7 = 10 becomes 3 + 7 = 10 10 = 10 which is a  true  statement, verifying that 3   is a solution to   x + 7 = 10

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Verify that - 6 is a solution to 5 y + 8 = 22 size 12{5y+8= - "22"} {}

When y = - 6 size 12{y=-6} {} ,

5 y + 8 = - 22 becomes 5 ( - 6 ) + 8 = - 22 - 30 + 8 = - 22 - 22 = - 22 which is a  true  statement, verifying that - is a solution to 5 y + 8 = - 22

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Verify that 5 is not a solution to a 1 = 2 a + 3 size 12{a - 1=2a+3} {} .

When a = 5 size 12{a=5} {} ,

a - 1 = 2 a + 3 becomes 5 - 1 = 2 5 + 3 5 - 1 = 10 + 3 4 = 13 a  false  statement, verifying that   5   is not a solution to  a - 1 = 2 a + 3

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Verify that -2 is a solution to 3 m 2 = 4 m 16 size 12{3m - 2= - 4m - "16"} {} .

When m = - 2 size 12{x=3} {} ,

3 m - 2 = - 4 m - 16 becomes 3 ( - 2 ) - 2 = - 4 ( - 2 ) - 16 - 6 - 2 = 8 - 16 - 8 = - 8 which is a   true  statement, verifying that - 2   is a solution to  3 m - 2 = - 4 m - 16

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Practice set a

Verify that 5 is a solution to m + 6 = 11 size 12{m+6="11"} {} .

Substitute 5 into m + 6 = 11 size 12{m+6="11"} {} . Does 5 plus 6 equal 11? Yes. Thus, 5 is a solution.

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Verify that - 5 is a solution to 2 m 4 = 14 size 12{2m - 4= - "14"} {} .

Substitute -5 into 2 m 4 = 14 size 12{2m - 4= - "14"} {} . does 2 time negative 5 minus 4 equal negative 14? Yes. Thus, -5 is a solution.

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Verify that 0 is a solution to 5 x + 1 = 1 size 12{5x+1=1} {} .

Substitute 0 into 5 x + 1 = 1 size 12{5x+1=1} {} . Does 5 times zero plus one equal 1? Yes. Thus, 0 is a solution.

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Verify that 3 is not a solution to 3 y + 1 = 4 y + 5 size 12{ - 3y+1=4y+5} {} .

Substitute 3 into 3 y + 1 = 4 y + 5 size 12{ - 3y+1=4y+5} {} . Does negative 3 times 3 plus 1 equal 4 times 3 plus 5? No. Thus, 3 is not a solution.

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Verify that -1 is a solution to 6 m 5 + 2 m = 7 m 6 size 12{6m - 5+2m=7m - 6} {} .

Substitute -1 into 6 m 5 + 2 m = 7 m 6 size 12{6m - 5+2m=7m - 6} {} . Does 6 times negative 1 minus 5 plus 2 times negative 1 equal 7 times negative 1 minus 6? Yes. Thus, -1 is a solution.

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Equivalent equations

Some equations have precisely the same collection of solutions. Such equations are called equivalent equations. For example, x - 5 = - 1 size 12{"x - 5 "=" -1"} {} , x + 7 = 11 size 12{"x "+" 7 "=" 11"} {} , and x = 4 size 12{x=4} {} are all equivalent equations since the only solution to each is x = 4 size 12{x=4} {} . (Can you verify this?)

Solving equations

We know 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.

This number is the same as this number
x size 12{x} {} = 4
x + 7 size 12{x+7} {} = 11
x 5 size 12{x - 5} {} = -1

Questions & Answers

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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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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.
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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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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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