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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 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
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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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