<< Chapter < Page Chapter >> Page >

Substitution and parallel lines

If computations eliminate all the variables and produce a contradiction, the two lines of a system are parallel, and the system is called inconsistent.

Sample set b

Solve the system { 2 x y = 1 4 x 2 y = 4 ( 1 ) ( 2 )

Step 1:  Solve equation 1 for y .
      2 x y = 1 y = 2 x + 1 y = 2 x 1

Step 2:  Substitute the expression 2 x 1 for y into equation 2.
      4 x 2 ( 2 x 1 ) = 4

Step 3:  Solve the equation obtained in step 2.
      4 x 2 ( 2 x 1 ) = 4 4 x 4 x + 2 = 4 2 4

Computations have eliminated all the variables and produce a contradiction. These lines are parallel.
A graph of two parallel lines. One line is labeled with the equation two x minus y is equal to one and passes through the points one, one, and zero, negative one. A second line is labeled with the equation four x minus two y is equal to four and passes through the points one, zero, and zero, negative two.
This system is inconsistent.

Got questions? Get instant answers now!

Practice set b

Slove the system { 7 x 3 y = 2 14 x 6 y = 1

Substitution produces 4 1 , or 1 2 2 , a contradiction. These lines are parallel and the system is inconsistent.

Got questions? Get instant answers now!

Substitution and coincident lines

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

Substitution and coincident lines

If computations eliminate all the variables and produce an identity, the two lines of a system are coincident and the system is called dependent.

Sample set c

Solve the system { 4 x + 8 y = 8 3 x + 6 y = 6 ( 1 ) ( 2 )

Step 1:  Divide equation 1 by 4 and solve for x .
      4 x + 8 y = 8 x + 2 y = 2 x = 2 y + 2

Step 2:  Substitute the expression 2 y + 2 for x in equation 2.
      3 ( 2 y + 2 ) + 6 y = 6

Step 3:  Solve the equation obtained in step 2.
      3 ( 2 y + 2 ) + 6 y = 6 6 y + 6 + 6 y = 6 6 = 6

Computations have eliminated all the variables and produced an identity. These lines are coincident.
A graph of two coincident lines. The line is labeled with the equation x plus two y is equal to two and a second label with the equation three x plus six y is equal to six. The lines pass through the points zero, one and two, zero. Since the lines are coincident, they have the same graph.
This system is dependent.

Got questions? Get instant answers now!

Practice set c

Solve the system { 4 x + 3 y = 1 8 x 6 y = 2

Computations produce 2 = 2 , an identity. These lines are coincident and the system is dependent.

Got questions? Get instant answers now!

Systems in which a coefficient of one of the variables is not 1 or cannot be made to be 1 without introducing fractions are not well suited for the substitution method. The problem in Sample Set D illustrates this “messy” situation.

Sample set d

Solve the system { 3 x + 2 y = 1 4 x 3 y = 3 ( 1 ) ( 2 )

Step 1:  We will solve equation ( 1 ) for y .
      3 x + 2 y = 1 2 y = 3 x + 1 y = 3 2 x + 1 2

Step 2:  Substitute the expression 3 2 x + 1 2 for y in equation ( 2 ) .
      4 x 3 ( 3 2 x + 1 2 ) = 3

Step 3:  Solve the equation obtained in step 2.
      4 x 3 ( 3 2 x + 1 2 ) = 3 Multiply both sides by the LCD ,  2 . 4 x + 9 2 x 3 2 = 3 8 x + 9 x 3 = 6 17 x 3 = 6 17 x = 9 x = 9 17

Step 4:  Substitute x = 9 17 into the equation obtained in step 1 , y = 3 2 x + 1 2 .
      y = 3 2 ( 9 17 ) + 1 2
      y = 27 34 + 17 34 = 10 34 = 5 17
     We now have x = 9 17 and y = 5 17 .

Step 5:  Substitution will show that these values of x and y check.

Step 6:  The solution is ( 9 17 , 5 17 ) .

Got questions? Get instant answers now!

Practice set d

Solve the system { 9 x 5 y = 4 2 x + 7 y = 9

These lines intersect at the point ( 1 , 1 ) .

Got questions? Get instant answers now!

Exercises

For the following problems, solve the systems by substitution.

{ 3 x + 2 y = 9 y = 3 x + 6

( 1 , 3 )

Got questions? Get instant answers now!

{ 5 x 3 y = 6 y = 4 x + 12

Got questions? Get instant answers now!

{ 2 x + 2 y = 0 x = 3 y 4

( 1 , 1 )

Got questions? Get instant answers now!

{ 3 x + 5 y = 9 x = 4 y 14

Got questions? Get instant answers now!

{ 3 x + y = 4 2 x + 3 y = 10

( 2 , 2 )

Got questions? Get instant answers now!

{ 4 x + y = 7 2 x + 5 y = 9

Got questions? Get instant answers now!

{ 6 x 6 = 18 x + 3 y = 3

( 4 , 1 3 )

Got questions? Get instant answers now!

{ x y = 5 2 x + y = 5

Got questions? Get instant answers now!

{ 5 x + y = 4 10 x 2 y = 8

Dependent (same line)

Got questions? Get instant answers now!

{ x + 4 y = 1 3 x 12 y = 1

Got questions? Get instant answers now!

{ 4 x 2 y = 8 6 x + 3 y = 0

( 1 , 2 )

Got questions? Get instant answers now!

{ 2 x + 3 y = 12 2 x + 4 y = 18

Got questions? Get instant answers now!

{ 3 x 9 y = 6 6 x 18 y = 5

inconsistent (parallel lines)

Got questions? Get instant answers now!

{ x + 4 y = 8 3 x 12 y = 10

Got questions? Get instant answers now!

{ x + y = 6 x y = 4

( 1 , 5 )

Got questions? Get instant answers now!

{ 2 x + y = 0 x 3 y = 0

Got questions? Get instant answers now!

{ 4 x 2 y = 7 y = 4

( 15 4 , 4 )

Got questions? Get instant answers now!

{ x + 6 y = 11 x = 1

Got questions? Get instant answers now!

{ 2 x 4 y = 10 3 x = 5 y + 12

( 1 , 3 )

Got questions? Get instant answers now!

{ y + 7 x + 4 = 0 x = 7 y + 28

Got questions? Get instant answers now!

{ x + 4 y = 0 x + 2 3 y = 10 3

( 4 , 1 )

Got questions? Get instant answers now!

{ x = 24 5 y x 5 4 y = 3 2

Got questions? Get instant answers now!

{ x = 11 6 y 3 x + 18 y = 33

inconsistent (parallel lines)

Got questions? Get instant answers now!

{ 2 x + 1 3 y = 4 3 x + 6 y = 39

Got questions? Get instant answers now!

{ 4 5 x + 1 2 y = 3 10 1 3 x + 1 2 y = 1 6

( 1 , 1 )

Got questions? Get instant answers now!

{ x 1 3 y = 8 3 3 x + y = 1

Got questions? Get instant answers now!

Exercises for review

( [link] ) Find the quotient: x 2 x 12 x 2 2 x 15 ÷ x 2 3 x 10 x 2 2 x 8 .

( x 4 ) 2 ( x 5 ) 2

Got questions? Get instant answers now!

( [link] ) Find the difference: x + 2 x 2 + 5 x + 6 x + 1 x 2 + 4 x + 3 .

Got questions? Get instant answers now!

( [link] ) Simplify 81 x 8 y 5 z 4 .

9 x 4 y 2 z 2 y

Got questions? Get instant answers now!

( [link] ) Use the quadratic formula to solve 2 x 2 + 2 x 3 = 0.

Got questions? Get instant answers now!

( [link] ) Solve by graphing { x y = 1 2 x + y = 5
An xy coordinate plane with gridlines labeled negative five and five with increments of one unit for both axes.

( 2 , 1 )
A graph of two lines intersecting at a point with coordinates negative two, one. One of the lines is passing through a point with coordinates zero, five and the other line is passing through a point with coordinates zero, negative one.

Got questions? Get instant answers now!

Questions & Answers

how to create a software using Android phone
Wiseman Reply
how
basra
what is the difference between C and C++.
Yan Reply
what is software
Sami Reply
software is a instructions like programs
Shambhu
what is the difference between C and C++.
Yan
yes, how?
Hayder
what is software engineering
Ahmad
software engineering is a the branch of computer science deals with the design,development, testing and maintenance of software applications.
Hayder
who is best bw software engineering and cyber security
Ahmad
Both software engineering and cybersecurity offer exciting career prospects, but your choice ultimately depends on your interests and skills. If you enjoy problem-solving, programming, and designing software syste
Hayder
what's software processes
Ntege Reply
I haven't started reading yet. by device (hardware) or for improving design Lol? Here. Requirement, Design, Implementation, Verification, Maintenance.
Vernon
I can give you a more valid answer by 5:00 By the way gm.
Vernon
it is all about designing,developing, testing, implementing and maintaining of software systems.
Ehenew
hello assalamualaikum
Sami
My name M Sami I m 2nd year student
Sami
what is the specific IDE for flutter programs?
Mwami Reply
jegudgdtgd my Name my Name is M and I have been talking about iey my papa john's university of washington post I tagged I will be in
Mwaqas Reply
yes
usman
how disign photo
atul Reply
hlo
Navya
hi
Michael
yes
Subhan
Show the necessary steps with description in resource monitoring process (CPU,memory,disk and network)
samuel Reply
What is software engineering
Tafadzwa Reply
Software engineering is a branch of computer science directed to writing programs to develop Softwares that can drive or enable the functionality of some hardwares like phone , automobile and others
kelvin
if any requirement engineer is gathering requirements from client and after getting he/she Analyze them this process is called
Alqa Reply
The following text is encoded in base 64. Ik5ldmVyIHRydXN0IGEgY29tcHV0ZXIgeW91IGNhbid0IHRocm93IG91dCBhIHdpbmRvdyIgLSBTdGV2ZSBXb3puaWFr Decode it, and paste the decoded text here
Julian Reply
what to do you mean
Vincent
hello
ALI
how are you ?
ALI
What is the command to list the contents of a directory in Unix and Unix-like operating systems
George Reply
how can i make my own software free of cost
Faizan Reply
like how
usman
hi
Hayder
The name of the author of our software engineering book is Ian Sommerville.
Doha Reply
what is software
Sampson Reply
the set of intruction given to the computer to perform a task
Noor
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