<< 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 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.

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

Substitution and parallel lines

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

Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
is it a question of log
Commplementary angles
Idrissa Reply
im all ears I need to learn
right! what he said ⤴⤴⤴
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
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
Got questions? Join the online conversation and get instant answers!
QuizOver.com 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?