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Substitution and parallel lines

If computations eliminate all the variables and produce a contradiction, the two lines of a system are parallel, and the system is called inconsistent.

Sample set b

Solve the system { 2 x y = 1 4 x 2 y = 4 ( 1 ) ( 2 )

Step 1:  Solve equation 1 for y .
      2 x y = 1 y = 2 x + 1 y = 2 x 1

Step 2:  Substitute the expression 2 x 1 for y into equation 2.
      4 x 2 ( 2 x 1 ) = 4

Step 3:  Solve the equation obtained in step 2.
      4 x 2 ( 2 x 1 ) = 4 4 x 4 x + 2 = 4 2 4

Computations have eliminated all the variables and produce a contradiction. These lines are parallel.
A graph of two parallel lines. One line is labeled with the equation two x minus y is equal to one and passes through the points one, one, and zero, negative one. A second line is labeled with the equation four x minus two y is equal to four and passes through the points one, zero, and zero, negative two.
This system is inconsistent.

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Practice set b

Slove the system { 7 x 3 y = 2 14 x 6 y = 1

Substitution produces 4 1 , or 1 2 2 , a contradiction. These lines are parallel and the system is inconsistent.

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Substitution and coincident lines

The following rule alerts us to the fact that the two lines of a system are coincident.

Substitution and coincident lines

If computations eliminate all the variables and produce an identity, the two lines of a system are coincident and the system is called dependent.

Sample set c

Solve the system { 4 x + 8 y = 8 3 x + 6 y = 6 ( 1 ) ( 2 )

Step 1:  Divide equation 1 by 4 and solve for x .
      4 x + 8 y = 8 x + 2 y = 2 x = 2 y + 2

Step 2:  Substitute the expression 2 y + 2 for x in equation 2.
      3 ( 2 y + 2 ) + 6 y = 6

Step 3:  Solve the equation obtained in step 2.
      3 ( 2 y + 2 ) + 6 y = 6 6 y + 6 + 6 y = 6 6 = 6

Computations have eliminated all the variables and produced an identity. These lines are coincident.
A graph of two coincident lines. The line is labeled with the equation x plus two y is equal to two and a second label with the equation three x plus six y is equal to six. The lines pass through the points zero, one and two, zero. Since the lines are coincident, they have the same graph.
This system is dependent.

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Practice set c

Solve the system { 4 x + 3 y = 1 8 x 6 y = 2

Computations produce 2 = 2 , an identity. These lines are coincident and the system is dependent.

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Systems in which a coefficient of one of the variables is not 1 or cannot be made to be 1 without introducing fractions are not well suited for the substitution method. The problem in Sample Set D illustrates this “messy” situation.

Sample set d

Solve the system { 3 x + 2 y = 1 4 x 3 y = 3 ( 1 ) ( 2 )

Step 1:  We will solve equation ( 1 ) for y .
      3 x + 2 y = 1 2 y = 3 x + 1 y = 3 2 x + 1 2

Step 2:  Substitute the expression 3 2 x + 1 2 for y in equation ( 2 ) .
      4 x 3 ( 3 2 x + 1 2 ) = 3

Step 3:  Solve the equation obtained in step 2.
      4 x 3 ( 3 2 x + 1 2 ) = 3 Multiply both sides by the LCD ,  2 . 4 x + 9 2 x 3 2 = 3 8 x + 9 x 3 = 6 17 x 3 = 6 17 x = 9 x = 9 17

Step 4:  Substitute x = 9 17 into the equation obtained in step 1 , y = 3 2 x + 1 2 .
      y = 3 2 ( 9 17 ) + 1 2
      y = 27 34 + 17 34 = 10 34 = 5 17
     We now have x = 9 17 and y = 5 17 .

Step 5:  Substitution will show that these values of x and y check.

Step 6:  The solution is ( 9 17 , 5 17 ) .

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Practice set d

Solve the system { 9 x 5 y = 4 2 x + 7 y = 9

These lines intersect at the point ( 1 , 1 ) .

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Exercises

For the following problems, solve the systems by substitution.

{ 3 x + 2 y = 9 y = 3 x + 6

( 1 , 3 )

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{ 5 x 3 y = 6 y = 4 x + 12

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{ 2 x + 2 y = 0 x = 3 y 4

( 1 , 1 )

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{ 3 x + 5 y = 9 x = 4 y 14

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{ 3 x + y = 4 2 x + 3 y = 10

( 2 , 2 )

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{ 4 x + y = 7 2 x + 5 y = 9

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{ 6 x 6 = 18 x + 3 y = 3

( 4 , 1 3 )

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{ x y = 5 2 x + y = 5

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{ 5 x + y = 4 10 x 2 y = 8

Dependent (same line)

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{ x + 4 y = 1 3 x 12 y = 1

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{ 4 x 2 y = 8 6 x + 3 y = 0

( 1 , 2 )

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{ 2 x + 3 y = 12 2 x + 4 y = 18

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{ 3 x 9 y = 6 6 x 18 y = 5

inconsistent (parallel lines)

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{ x + 4 y = 8 3 x 12 y = 10

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{ x + y = 6 x y = 4

( 1 , 5 )

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{ 2 x + y = 0 x 3 y = 0

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{ 4 x 2 y = 7 y = 4

( 15 4 , 4 )

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{ x + 6 y = 11 x = 1

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{ 2 x 4 y = 10 3 x = 5 y + 12

( 1 , 3 )

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{ y + 7 x + 4 = 0 x = 7 y + 28

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{ x + 4 y = 0 x + 2 3 y = 10 3

( 4 , 1 )

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{ x = 24 5 y x 5 4 y = 3 2

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{ x = 11 6 y 3 x + 18 y = 33

inconsistent (parallel lines)

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{ 2 x + 1 3 y = 4 3 x + 6 y = 39

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{ 4 5 x + 1 2 y = 3 10 1 3 x + 1 2 y = 1 6

( 1 , 1 )

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{ x 1 3 y = 8 3 3 x + y = 1

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Exercises for review

( [link] ) Find the quotient: x 2 x 12 x 2 2 x 15 ÷ x 2 3 x 10 x 2 2 x 8 .

( x 4 ) 2 ( x 5 ) 2

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( [link] ) Find the difference: x + 2 x 2 + 5 x + 6 x + 1 x 2 + 4 x + 3 .

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( [link] ) Simplify 81 x 8 y 5 z 4 .

9 x 4 y 2 z 2 y

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( [link] ) Use the quadratic formula to solve 2 x 2 + 2 x 3 = 0.

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( [link] ) Solve by graphing { x y = 1 2 x + y = 5
An xy coordinate plane with gridlines labeled negative five and five with increments of one unit for both axes.

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

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Questions & Answers

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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
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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