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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

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
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 .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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.
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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