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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: know the properties used in the addition method, be able to use the addition method to solve a system of linear equations, know what to expect when using the addition method with a system that consists of parallel or coincident lines.

Overview

  • The Properties Used in the Addition Method
  • The Addition Method
  • Addition and Parallel or Coincident Lines

The properties used in the addition method

Another method of solving a system of two linear equations in two variables is called the method of elimination by addition . It is similar to the method of elimination by substitution in that the process eliminates one equation and one variable. The method of elimination by addition makes use of the following two properties.

  1. If A , B , and C are algebraic expressions such that

    A = B C = D A + C = B + D and then
  2. a x + ( a x ) = 0


Property 1 states that if we add the left sides of two equations together and the right sides of the same two equations together, the resulting sums will be equal. We call this adding equations . Property 2 states that the sum of two opposites is zero.

The addition method

To solve a system of two linear equations in two variables by 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. Do one of the following:
     (a)  Substitute the value obtained in step 4 into either of the original equations and solve to obtain the value of the other variable,
     or
     (b)  Repeat steps 1-5 for the other variable.
  6. Check the solutions in both equations.
  7. Write the solution as an ordered pair.


The addition method works well when the coefficient of one of the variables is 1 or a number other than 1.

Sample set a

Solve  { x y = 2 ( 1 ) 3 x + y = 14 ( 2 )

Step 1:  Both equations appear in the proper form.

Step 2:  The coefficients of y are already opposites, 1 and 1 , so there is no need for a multiplication.

Step 3:  Add the equations.

      x y = 2 3 x + y = 14 4 x + 0 = 16

Step 4:  Solve the equation 4 x = 16.

      4 x = 16

      x = 4

 The problem is not solved yet; we still need the value of y .

Step 5:  Substitute x = 4 into either of the original equations. We will use equation 1.

      4 y = 2 Solve for  y . y = 2 y = 2

 We now have x = 4 , y = 2.

Step 6:  Substitute x = 4 and y = 2 into both the original equations for a check.

       ( 1 ) x y = 2 ( 2 ) 3 x + y = 14 4 2 = 2 Is this correct? 3 ( 4 ) + 2 = 14 Is this correct? 2 = 2 Yes, this is correct . 12 + 2 = 14 Is this correct? 14 = 14 Yes, this is correct .

Step 7:  The solution is ( 4 , 2 ) .

The two lines of this system intersect at ( 4 , 2 ) .

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
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Stoney Reply
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Kyle
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biomolecules are e building blocks of every organics and inorganic materials.
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research.net
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sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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Do somebody tell me a best nano engineering book for beginners?
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there is no specific books for beginners but there is book called principle of nanotechnology
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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
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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.
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for screen printed electrodes ?
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of graphene you mean?
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or in general
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in general
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Graphene has a hexagonal structure
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Source:  OpenStax, Algebra i for the community college. OpenStax CNX. Dec 19, 2014 Download for free at http://legacy.cnx.org/content/col11598/1.3
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