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Practice set a

Solve each system by addition.

{ x + y = 6 2 x y = 0

( 2 , 4 )

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

( 4 , 2 )

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Sample set b

Solve the following systems using the addition method.

Solve { 6 a 5 b = 14 ( 1 ) 2 a + 2 b = 10 ( 2 )

Step 1: The equations are already in the proper form, a x + b y = c .

Step 2: If we multiply equation (2) by —3, the coefficients of a will be opposites and become 0 upon addition, thus eliminating a .

       { 6 a 5 b = 14 3 ( 2 a + 2 b ) = 3 ( 10 ) { 6 a 5 b = 14 6 a 6 b = 30

Step 3:  Add the equations.

       6 a 5 b = 14 6 a 6 b = 30 0 11 b = 44

Step 4:  Solve the equation 11 b = 44.

       11 b = 44
        b = 4

Step 5:  Substitute b = 4 into either of the original equations. We will use equation 2.

       2 a + 2 b = 10 2 a + 2 ( 4 ) = 10 Solve for  a . 2 a 8 = 10 2 a = 2 a = 1

 We now have a = 1 and b = 4.

Step 6:  Substitute a = 1 and b = 4 into both the original equations for a check.

       ( 1 ) 6 a 5 b = 14 ( 2 ) 2 a + 2 b = 10 6 ( 1 ) 5 ( 4 ) = 14 Is this correct? 2 ( 1 ) + 2 ( 4 ) = 10 Is this correct? 6 + 20 = 14 Is this correct? 2 8 = 10 Is this correct? 14 = 14 Yes, this is correct . 10 = 10 Yes, this is correct .

Step 7:  The solution is ( 1 , 4 ) .

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Solve  { 3 x + 2 y = 4 4 x = 5 y + 10 ( 1 ) ( 2 )

Step 1:  Rewrite the system in the proper form.

       { 3 x + 2 y = 4 4 x 5 y = 10 ( 1 ) ( 2 )

Step 2:  Since the coefficients of y already have opposite signs, we will eliminate y .
     Multiply equation (1) by 5, the coefficient of y in equation 2.
     Multiply equation (2) by 2, the coefficient of y in equation 1.

       { 5 ( 3 x + 2 y ) = 5 ( 4 ) 2 ( 4 x 5 y ) = 2 ( 10 ) { 15 x + 10 y = 20 8 x 10 y = 20

Step 3:  Add the equations.

       15 x + 10 y = 20 8 x 10 y = 20 23 x + 0 = 0

Step 4:  Solve the equation 23 x = 0

       23 x = 0

       x = 0

Step 5:  Substitute x = 0 into either of the original equations. We will use equation 1.

       3 x + 2 y = 4 3 ( 0 ) + 2 y = 4 Solve for  y . 0 + 2 y = 4 y = 2

 We now have x = 0 and y = 2.

Step 6:  Substitution will show that these values check.

Step 7:  The solution is ( 0 , 2 ) .

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Practice set b

Solve each of the following systems using the addition method.

{ 3 x + y = 1 5 x + y = 3

( 1 , 2 )

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{ x + 4 y = 1 x 2 y = 5

( 3 , 1 )

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

( 2 , 2 )

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{ 5 x 3 y = 1 8 x 6 y = 4

( 1 , 2 )

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{ 3 x 5 y = 9 4 x + 8 y = 12

( 3 , 0 )

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Addition and parallel or coincident lines

When the lines of a system are parallel or coincident, the method of elimination produces results identical to that of the method of elimination by substitution.

Addition and parallel lines

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

Addition and coincident lines

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

Sample set c

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

Step 1: The equations are in the proper form.

Step 2: We can eliminate x by multiplying equation (1) by –2.

       { 2 ( 2 x y ) = 2 ( 1 ) 4 x 2 y = 4 { 4 x + 2 y = 2 4 x 2 y = 4

Step 3:  Add the equations.

       4 x + 2 y = 2 4 x 2 y = 4 0 + 0 = 2 0 = 2

 This is false and is therefore a contradiction. The lines of this system are parallel.  This system is inconsistent.

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

Step 1:  The equations are in the proper form.

Step 2:  We can eliminate x by multiplying equation (1) by –3 and equation (2) by 4.

       { 3 ( 4 x + 8 y ) = 3 ( 8 ) 4 ( 3 x + 6 y ) = 4 ( 6 ) { 12 x 24 y = 24 12 x + 24 y = 24

Step 3:  Add the equations.

       12 x 24 y = 24 12 x + 24 y = 24 0 + 0 = 0 0 = 0

 This is true and is an identity. The lines of this system are coincident.

 This system is dependent.

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Practice set c

Solve each of the following systems using the addition method.

{ x + 2 y = 6 6 x + 12 y = 1

inconsistent

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{ 4 x 28 y = 4 x 7 y = 1

dependent

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Exercises

For the following problems, solve the systems using elimination by addition.

{ x + y = 11 x y = 1

( 5 , 6 )

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{ x + 3 y = 13 x 3 y = 11

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

( 2 , 2 )

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{ 2 x 7 y = 1 5 x + 7 y = 22

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{ 3 x + 4 y = 24 3 x 7 y = 42

( 0 , 6 )

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{ 8 x + 5 y = 3 9 x 5 y = 71

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

( 2 , 2 )

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

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{ x + y = 4 x y = 4

dependent

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

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{ 3 x + 4 y = 7 x + 5 y = 6

( 1 , 1 )

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{ 4 x 2 y = 2 7 x + 4 y = 26

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{ 3 x + y = 4 5 x 2 y = 14

( 2 , 2 )

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{ 5 x 3 y = 20 x + 6 y = 4

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{ 6 x + 2 y = 18 x + 5 y = 19

( 4 , 3 )

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{ x 11 y = 17 2 x 22 y = 4

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{ 2 x + 3 y = 20 3 x + 2 y = 15

( 1 , 6 )

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

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{ 3 x 4 y = 2 9 x 12 y = 6

dependent

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{ 3 x 5 y = 28 4 x 2 y = 20

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{ 6 x 3 y = 3 10 x 7 y = 3

( 1 , 1 )

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{ 4 x + 12 y = 0 8 x + 16 y = 0

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{ 3 x + y = 1 12 x + 4 y = 6

inconsistent

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{ 8 x + 5 y = 23 3 x 3 y = 12

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{ 2 x + 8 y = 10 3 x + 12 y = 15

dependent

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{ 4 x + 6 y = 8 6 x + 8 y = 12

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{ 10 x + 2 y = 2 15 x 3 y = 3

inconsistent

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{ x + 3 4 y = 1 2 3 5 x + y = 7 5

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

( 0 , 4 )

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{ 8 x 3 y = 25 4 x 5 y = 5

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{ 10 x 4 y = 72 9 x + 5 y = 39

( 258 7 , 519 7 )

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{ 12 x + 16 y = 36 10 x + 12 y = 30

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{ 25 x 32 y = 14 50 x + 64 y = 28

dependent

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Exercises for review

( [link] ) Simplify and write ( 2 x 3 y 4 ) 5 ( 2 x y 6 ) 5 so that only positive exponents appear.

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( [link] ) Simplify 8 + 3 50 .

17 2

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( [link] ) Solve the radical equation 2 x + 3 + 5 = 8.

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( [link] ) Solve by graphing { x + y = 4 3 x y = 0
An xy coordinate plane with gridlines labeled negative five and five with increments of one unit for both axes.

( 1 , 3 )
A graph of two lines intersecting at a point with coordinates negative one, three. 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, four and four, zero.

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( [link] ) Solve using the substitution method: { 3 x 4 y = 11 5 x + y = 3

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Questions & Answers

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
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Abigail
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Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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
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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Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
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Ramkumar Reply
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AMJAD
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Victor 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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