<< 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

Ayele, K., 2003. Introductory Economics, 3rd ed., Addis Ababa.
Widad Reply
can you send the book attached ?
Ariel
?
Ariel
What is economics
Widad Reply
the study of how humans make choices under conditions of scarcity
AI-Robot
U(x,y) = (x×y)1/2 find mu of x for y
Desalegn Reply
U(x,y) = (x×y)1/2 find mu of x for y
Desalegn
what is ecnomics
Jan Reply
this is the study of how the society manages it's scarce resources
Belonwu
what is macroeconomic
John Reply
macroeconomic is the branch of economics which studies actions, scale, activities and behaviour of the aggregate economy as a whole.
husaini
etc
husaini
difference between firm and industry
husaini Reply
what's the difference between a firm and an industry
Abdul
firm is the unit which transform inputs to output where as industry contain combination of firms with similar production 😅😅
Abdulraufu
Suppose the demand function that a firm faces shifted from Qd  120 3P to Qd  90  3P and the supply function has shifted from QS  20  2P to QS 10  2P . a) Find the effect of this change on price and quantity. b) Which of the changes in demand and supply is higher?
Toofiq Reply
explain standard reason why economic is a science
innocent Reply
factors influencing supply
Petrus Reply
what is economic.
Milan Reply
scares means__________________ends resources. unlimited
Jan
economics is a science that studies human behaviour as a relationship b/w ends and scares means which have alternative uses
Jan
calculate the profit maximizing for demand and supply
Zarshad Reply
Why qualify 28 supplies
Milan
what are explicit costs
Nomsa Reply
out-of-pocket costs for a firm, for example, payments for wages and salaries, rent, or materials
AI-Robot
concepts of supply in microeconomics
David Reply
economic overview notes
Amahle Reply
identify a demand and a supply curve
Salome Reply
i don't know
Parul
there's a difference
Aryan
Demand curve shows that how supply and others conditions affect on demand of a particular thing and what percent demand increase whith increase of supply of goods
Israr
Hi Sir please how do u calculate Cross elastic demand and income elastic demand?
Abari
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