<< Chapter < Page Chapter >> Page >
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} {} .

Got questions? Get instant answers now!

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"} {}

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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} {} .

Got questions? Get instant answers now!

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} {}

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

Questions & Answers

what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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 did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
7hours 36 min - 4hours 50 min
Tanis Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Fundamentals of mathematics' conversation and receive update notifications?

Ask