<< Chapter < Page Chapter >> Page >
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.


  • 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,
     (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 ) .

Got questions? Get instant answers now!

Questions & Answers

Application of nanotechnology in medicine
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.
yes that's correct
I think
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Please keep in mind that it's not allowed to promote any social groups (whatsapp, facebook, etc...), exchange phone numbers, email addresses or ask for personal information on QuizOver's platform.
QuizOver Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?