<< 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.

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 ) .

Got questions? Get instant answers now!

Questions & Answers

what is mutation
Janga Reply
what is a cell
Sifune Reply
how is urine form
Sifune
what is antagonism?
mahase Reply
classification of plants, gymnosperm features.
Linsy Reply
what is the features of gymnosperm
Linsy
how many types of solid did we have
Samuel Reply
what is an ionic bond
Samuel
What is Atoms
Daprince Reply
what is fallopian tube
Merolyn
what is bladder
Merolyn
what's bulbourethral gland
Eduek Reply
urine is formed in the nephron of the renal medulla in the kidney. It starts from filtration, then selective reabsorption and finally secretion
onuoha Reply
State the evolution relation and relevance between endoplasmic reticulum and cytoskeleton as it relates to cell.
Jeremiah
what is heart
Konadu Reply
how is urine formed in human
Konadu
how is urine formed in human
Rahma
what is the diference between a cavity and a canal
Pelagie Reply
what is the causative agent of malaria
Diamond
malaria is caused by an insect called mosquito.
Naomi
Malaria is cause by female anopheles mosquito
Isaac
Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
Olalekan
a canal is more needed in a root but a cavity is a bad effect
Commander
what are pathogens
Don Reply
In biology, a pathogen (Greek: πάθος pathos "suffering", "passion" and -γενής -genēs "producer of") in the oldest and broadest sense, is anything that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a germ. The term pathogen came into use in the 1880s.[1][2
Zainab
A virus
Commander
Definition of respiration
Muhsin Reply
respiration is the process in which we breath in oxygen and breath out carbon dioxide
Achor
how are lungs work
Commander
where does digestion begins
Achiri Reply
in the mouth
EZEKIEL
what are the functions of follicle stimulating harmones?
Rashima Reply
stimulates the follicle to release the mature ovum into the oviduct
Davonte
what are the functions of Endocrine and pituitary gland
Chinaza
endocrine secrete hormone and regulate body process
Achor
while pituitary gland is an example of endocrine system and it's found in the Brain
Achor
what's biology?
Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
Lisah
biology is the study of life.
Alfreda
Biology is the study of how living organisms live and survive in a specific environment
Sifune
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




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?

Ask