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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. Beginning with the graphical solution of systems, this chapter includes an interpretation of independent, inconsistent, and dependent systems and examples to illustrate the applications for these systems. The substitution method and the addition method of solving a system by elimination are explained, noting when to use each method. The five-step method is again used to illustrate the solutions of value and rate problems (coin and mixture problems), using drawings that correspond to the actual situation.This module presents a summary of the key concepts of the chapter "Systems of Linear Equations".

Summary of key concepts

System of equations ( [link] )

A collection of two linear equations in two variables is called a system of equations.

Solution to a system ( [link] )

An ordered pair that is a solution to both equations in a system is called a solution to the system of equations. The values x = 3 , y = 1 are a solution to the system

{ x y = 2 x + y = 4

Independent systems ( [link] )

Systems in which the lines intersect at precisely one point are independent systems. In applications, independent systems can arise when the collected data are accurate and complete.

Inconsistent systems ( [link] )

Systems in which the lines are parallel are inconsistent systems. In applications, inconsistent systems can arise when the collected data are contradictory.

Dependent systems ( [link] )

Systems in which the lines are coincident (one on the other) are dependent systems. In applications, dependent systems can arise when the collected data are incomplete.

Solving a system by graphing ( [link] )

To solve a system by graphing:
  1. Graph each equation of the same set of axes.
  2. If the lines intersect, the solution is the point of intersection.

Solving a system by substitution ( [link] )

To solve a system using substitution,
  1. Solve one of the equations for one of the variables.
  2. Substitute the expression for the variable chosen in step 1 into the other equation.
  3. Solve the resulting equation in one variable.
  4. Substitute the value obtained in step 3 into the equation obtained in step 1 and solve to obtain the value of the other variable.
  5. Check the solution in both equations.
  6. Write the solution as an ordered pair.

Solving a system by addition ( [link] )

To solve a system using addition,
  1. Write, if necessary, both equations in general form

    a x + b y = c
  2. If necessary, multiply one or both equations by factors that will produce opposite coefficients for one of the variables.
  3. Add the equations to eliminate one equation and one variable.
  4. Solve the equation obtained in step 3.
  5. Substitute the value obtained in step 4 into either of the original equations and solve to obtain the value of the other variable.
  6. Check the solution in both equations.
  7. Write the solution as an ordered pair.

Substitution and addition and parallel lines ( [link] , [link] )

If computations eliminate all variables and produce a contradiction, the two lines of the system are parallel and no solution exists. The system is inconsistent.

Substitution and addition and coincident lines ( [link] , [link] )

If computations eliminate all variables and produce an identity, the two lines of the system are coincident and the system has infinitely many solutions. The system is dependent.

Applications ( [link] )

The five-step method can be used to solve applied problems that involve linear systems that consist of two equations in two variables. The solutions of number problems, mixture problems, and value and rate problems are examined in this section. The rate problems have particular use in chemistry.

Questions & Answers

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 to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
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
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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