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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.Objectives of this module: be able to recognize a system of equations and a solution to it, be able to graphically interpret independent, inconsistent, and dependent systems, be able to solve a system of linear equations graphically.

Overview

  • Systems of Equations
  • Solution to A System of Equations
  • Graphs of Systems of Equations
  • Independent, Inconsistent, and Dependent Systems
  • The Method of Solving A System Graphically

Systems of equations

Systems of equations

A collection of two linear equations in two variables is called a system of linear equations in two variables , or more briefly, a system of equations . The pair of equations

{ 5 x 2 y = 5 x + y = 8
is a system of equations. The brace { is used to denote that the two equations occur together (simultaneously).

Solution to a system of equations

Solution to a system

We know that one of the infinitely many solutions to one linear equation in two variables is an ordered pair. An ordered pair that is a solution to both of the equations in a system is called a solution to the system of equations . For example, the ordered pair ( 3 , 5 ) is a solution to the system

{ 5 x 2 y = 5 x + y = 8
since ( 3 , 5 ) is a solution to both equations.

5 x 2 y = 5 x + y = 8 5 ( 3 ) 2 ( 5 ) = 5 Is this correct? 3 + 5 = 8 Is this correct? 15 10 = 5 Is this correct? 8 = 8 Yes, this is correct . 5 = 5 Yes, this is correct .

Graphs of systems of equations

One method of solving a system of equations is by graphing. We know that the graph of a linear equation in two variables is a straight line. The graph of a system will consist of two straight lines. When two straight lines are graphed, one of three possibilities may result.

The lines intersect at the point ( a , b ) . The point ( a , b ) is the solution to the corresponding system.
A graph of two lines; 'line one' and 'line two,' intersecting at a point labeled with coordinates (a, b) and with a second label with x-coordinate negative one and one-half, and y-coordinate negative one and one-half. Line one is passing through a point with coordinates zero, one over two, and line two is passing through a point with coordinates negative four and one half, zero.

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The lines are parallel. They do not intersect. The system has no solution.
A graph of two parallel lines; 'Line one' and 'Line two'. Line one is passing through two points with the coordinates zero, one, and five, negative two. Line two is passing through two points with the coordinates zero, three, and five, zero.

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The lines are coincident (one on the other). They intersect at infinitely many points. The system has infinitely many solutions.
A graph of two conincident lines; 'Line one' and 'Line two'. The lines are passsing through the same two points with the coordinates negative three, negative one, and four, three. Since the lines are coincident lines they have the same graph.

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Independent, inconsistent, and dependent systems

Independent systems

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

The sum of two numbers is 10 and the product of the two numbers is 21. Find the numbers.

In this application, the data are accurate and complete. The solution is 7 and 3.

Inconsistent systems

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

The sum of two even numbers is 30 and the difference of the same two numbers is 0. Find the numbers.

The data are contradictory. There is no solution to this application.

Dependent systems

Systems in which the lines are coincident are called dependent systems . In applications, dependent systems can arise when the collected data are incomplete. For example.

The difference of two numbers is 9 and twice one number is 18 more than twice the other.

The data are incomplete. There are infinitely many solutions.

The method of solving a system graphically

The method of solving a system graphically

To solve a system of equations graphically: Graph both equations.
  1. If the lines intersect, the solution is the ordered pair that corresponds to the point of intersection. The system is independent.
  2. If the lines are parallel, there is no solution. The system is inconsistent.
  3. If the lines are coincident, there are infinitely many solutions. The system is dependent.

Sample set a

Solve each of the following systems by graphing.

{ 2 x + y = 5 x + y = 2 ( 1 ) ( 2 )
Write each equation in slope-intercept form.

( 1 ) 2 x + y = 5 ( 2 ) x + y = 2 y = 2 x + 5 y = x + 2
Graph each of these equations.
A graph of two lines; ‘one’ and ‘two.’ The lines are intersecting at a point with coordinates negative one, three. Line one is passing through a point with coordinates zero, five. Line two is passing through two points with coordinates zero, two, and one, one.
The lines appear to intersect at the point ( 1 , 3 ) . The solution to this system is ( 1 , 3 ) , or

x = 1 , y = 3

Check:   Substitute x = 1 , y = 3 into each equation.

( 1 ) 2 x + y = 5 ( 2 ) x + y = 2 2 ( 1 ) + 3 = 5 Is this correct? 1 + 3 = 2 Is this correct? 2 + 3 = 5 Is this correct? 2 = 2 Yes, this is correct . 5 = 5 Yes, this is correct .

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{ x + y = 1 x + y = 2 ( 1 ) ( 2 )
Write each equation in slope-intercept form.

( 1 ) x + y = 1 ( 2 ) x + y = 2 y = x 1 y = x + 2
Graph each of these equations.
A graph of two parallel line; 'one' and 'two'. Line one is passing through two points with the coordinates zero, two, and one, three. Line two is passing through two points with the coordinates zero, negative one, and one, zero.
These lines are parallel. This system has no solution. We denote this fact by writing inconsistent .

We are sure that these lines are parallel because we notice that they have the same slope, m = 1 for both lines. The lines are not coincident because the y -intercepts are different.

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{ 2 x + 3 y = 2 6 x + 9 y = 6 ( 1 ) ( 2 )
Write each equation in slope-intercept form.

( 1 ) 2 x + 3 y = 2 ( 2 ) 6 x + 9 y = 6 3 y = 2 x 2 9 y = 6 x 6 y = 2 3 x 2 3 y = 2 3 x 2 3
A graph of two conincident lines; 'one' and 'two'. The lines are passsing through the same two points with the coordinates zero, negative two over three, and three, one and one third. Since the lines are coincident, they have the same graph.
Both equations are the same. This system has infinitely many solutions. We write dependent .

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Practice set a

Solve each of the following systems by graphing. Write the ordered pair solution or state that the system is inconsistent, or dependent.

{ 2 x + y = 1 x + y = 5
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

x = 2 , y = 3
A graph of two lines intersecting at a point with the coordinates two, negative three. One of the lines is passing through a point with  the coordinates one over two, zero, and the other line is passing through a point with the coordinates zero, negative five.

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{ 2 x + 3 y = 6 6 x 9 y = 18
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

dependent
A graph of two coincident lines passing through the same two points with the coordinates zero, two, and three, four. Since the lines are coincident, they have the same graph. The graph is labeled as 'coincident lines.'

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{ 3 x + 5 y = 15 9 x + 15 y = 15
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

inconsistent
A A graph of two parallel lines. One of the lines is passing through two points with coordinates zero, one and one and two third, zero. The other line is passing through two points with coordinates zero, three, and five, zero. The graph is labeled as 'parallel lines.'

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{ y = 3 x + 2 y = 4
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

x = 2 , y = 3
A graph of two lines intersecting at a point with the coordinates two, negative three. One of the lines is passing through a point with  the coordinates one zero, negative two. The other line is parallel to x axis, and is passing  through a point with the coordinates negative three, negative three.

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Exercises

For the following problems, solve the systems by graphing. Write the ordered pair solution, or state that the system is inconsistent or dependent.

{ x + y = 5 x + y = 1
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

( 3 , 2 )
A graph of two lines intersecting at a point with coordinates negative three, negative two. One of the lines is passing through a point with coordinates zero, negative five and, the other line is passing through two points with coordinates negative one, zero; and zero, one.

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{ 3 x + y = 5 x + y = 3
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

( 1 , 2 )
A graph of two lines intersecting at a point with coordinates negative one, two. One of the lines is passing through a point with coordinates zero, five, and the other line is passing through two points with coordinates zero, three; and one, four.

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{ x y = 6 x + 2 y = 0
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

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{ 3 x + y = 0 4 x 3 y = 12
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

( 12 13 , 36 13 )
A graph of two lines intersecting at a point with coordinates twelve over thirteen, negative thirty-six over thirteen. One of the lines is passing through a point with coordinates zero, zero and the other line is passing through two points with coordinates zero, negative four; and three, zero.

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{ 4 x + y = 7 3 x + y = 2
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

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{ 2 x + 3 y = 6 3 x + 4 y = 6
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

These coordinates are hard to estimate. This problem illustrates that the graphical method is not always the most accurate. ( 6 , 6 )
A graph of two lines intersecting at a point with coordinates negative six, six. One of the lines is passing through a point with coordinates zero, three over two and the other line is passing through two points with coordinates zero, two; and three, zero.

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{ x + y = 3 4 x + 4 y = 12
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

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{ 2 x 3 y = 1 4 x 6 y = 4
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

inconsistent
A graph of two parallel lines. One of the lines is passing through two points with coordinates zero, negative two over three and three, zero. The other line is passing through a point with coordinates zero, negative one over three.

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{ x + 2 y = 3 3 x 6 y = 9
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

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{ x 2 y = 6 3 x 6 y = 18
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

dependent
A graph of two coincident lines passing through the same two points with coordinates zero, negative three; and two, negative two. Since the lines are coincident, they have the same graph.

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{ 2 x + 3 y = 6 10 x 15 y = 30
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

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Exercises for review

( [link] ) Express 0.000426 in scientific notation.

4.26 × 10 4

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( [link] ) Find the product: ( 7 x 3 ) 2 .

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( [link] ) Supply the missing word. The of a line is a measure of the steepness of the line.

slope

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( [link] ) Supply the missing word. An equation of the form a x 2 + b x + c = 0 , a 0 , is called a equation.

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( [link] ) Construct the graph of the quadratic equation y = x 2 3.
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

A graph of a parabola passing through four points with coordinates negative two, one; negative one, negative two; one, negative two; and two, one.

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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
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
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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