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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 when the substitution method works best, be able to use the substitution method to solve a system of linear equations, know what to expect when using substitution with a system that consists of parallel lines.


  • When Substitution Works Best
  • The Substitution Method
  • Substitution and Parallel Lines
  • Substitution and Coincident Lines

When substitution works best

We know how to solve a linear equation in one variable. We shall now study a method for solving a system of two linear equations in two variables by transforming the two equations in two variables into one equation in one variable.

To make this transformation, we need to eliminate one equation and one variable. We can make this elimination by substitution .

When substitution works best

The substitution method works best when either of these conditions exists:
  1. One of the variables has a coefficient of 1 , or
  2. One of the variables can be made to have a coefficient of 1 without introducing fractions.

The substitution method

The substitution method

To solve a system of two linear equations in two variables,
  1. Solve one of the equations for one of the variables.
  2. Substitute the expression for the variable chosen in step 1 into the other equation.
  3. Solve the resulting equation in one variable.
  4. Substitute the value obtained in step 3 into the equation obtained in step 1 and solve to obtain the value of the other variable.
  5. Check the solution in both equations.
  6. Write the solution as an ordered pair.

Sample set a

Solve the system { 2 x + 3 y = 14 3 x + y = 7 ( 1 ) ( 2 )

Step 1:  Since the coefficient of y in equation 2 is 1, we will solve equation 2 for y .

       y = 3 x + 7

Step 2:  Substitute the expression 3 x + 7 for y in equation 1.

       2 x + 3 ( 3 x + 7 ) = 14

Step 3:  Solve the equation obtained in step 2.
      2 x + 3 ( 3 x + 7 ) = 14 2 x 9 x + 21 = 14 7 x + 21 = 14 7 x = 7 x = 1
Step 4:  Substitute x = 1 into the equation obtained in step 1 , y = 3 x + 7.
      y = 3 ( 1 ) + 7 y = 3 + 7 y = 4
 We now have x = 1 and y = 4.

Step 5:  Substitute x = 1 , y = 4 into each of the original equations for a check.
( 1 ) 2 x + 3 y = 14 ( 2 ) 3 x + y = 7 2 ( 1 ) + 3 ( 4 ) = 14 Is this correct? 3 ( 1 ) + ( 4 ) = 7 Is this correct? 2 + 12 = 14 Is this correct? 3 + 4 = 7 Is this correct? 14 = 14 Yes, this is correct . 7 = 7 Yes, this is correct .

Step 6:  The solution is ( 1 , 4 ) . The point ( 1 , 4 ) is the point of intersection of the two lines of the system.

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Practice set a

Slove the system { 5 x 8 y = 18 4 x + y = 7

The point ( 2 , 1 ) is the point of intersection of the two lines.

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Substitution and parallel lines

The following rule alerts us to the fact that the two lines of a system are parallel.

Questions & Answers

How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
How can I make nanorobot?
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
how can I make nanorobot?
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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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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