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These are only three of the infintely many solutions to this equation.

Sample set a

Find a solution to each of the following linear equations in two variables and write the solution as an ordered pair.

y = 3 x 6 , if x = 1

Substitute 1 for x , compute, and solve for y .

y = 3 ( 1 ) 6 = 3 6 = 3

Hence, one solution is ( 1 , 3 ) .

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y = 15 4 x , if x = 10

Substitute 10 for x , compute, and solve for y .

y = 15 4 ( 10 ) = 15 + 40 = 55

Hence, one solution is ( 10 , 55 ) .

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b = 9 a + 21 , if a = 2

Substitute 2 for a , compute, and solve for b .

b = 9 ( 2 ) + 21 = 18 + 21 = 3

Hence, one solution is ( 2 , 3 ) .

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5 x 2 y = 1 , if x = 0

Substitute 0 for x , compute, and solve for y .

5 ( 0 ) 2 y = 1 0 2 y = 1 2 y = 1 y = 1 2

Hence, one solution is ( 0 , 1 2 ) .

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Practice set a

Find a solution to each of the following linear equations in two variables and write the solution as an ordered pair.

y = 7 x 20 , if x = 3

( 3 , 1 )

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m = 6 n + 1 , if n = 2

( 2 , 11 )

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b = 3 a 7 , if a = 0

( 0 , 7 )

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10 x 5 y 20 = 0 , if x = 8

( 8 , 20 )

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3 a + 2 b + 6 = 0 , if a = 1

( 1 , 3 2 )

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Exercises

For the following problems, solve the linear equations in two variables.

y = 8 x + 14 , if x = 1

( 1 , 22 )

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y = 2 x + 1 , if x = 0

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y = 5 x + 6 , if x = 4

( 4 , 26 )

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x + y = 7 , if x = 8

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3 x + 4 y = 0 , if x = 3

( 3 , 9 4 )

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2 x + y = 1 , if x = 1 2

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5 x 3 y + 1 = 0 , if x = 6

( 6 , 29 3 )

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4 x 4 y = 4 , if y = 7

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2 x + 6 y = 1 , if y = 0

( 1 2 , 0 )

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x y = 0 , if y = 14 3

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y = x , if x = 1

( 1 , 1 )

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x + y = 0 , if x = 0

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y + 3 4 = x , if x = 9 4

( 9 4 , 3 2 )

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y + 17 = x , if x = 12

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20 y + 14 x = 1 , if x = 8

( 8 , 111 20 )

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3 5 y + 1 4 x = 1 2 , if x = 3

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1 5 x + y = 9 , if y = 1

( 40 , 1 )

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y + 7 x = 0 , if x = A star

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2 x + 31 y 3 = 0 , if x = a

( a , 3 2 a 31 )

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436 x + 189 y = 881 , if x = 4231

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y = 6 ( x 7 ) , if x = 2

( 2 , 30 )

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y = 2 ( 4 x + 5 ) , if x = 1

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5 y = 9 ( x 3 ) , if x = 2

( 2 , 9 5 )

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3 y = 4 ( 4 x + 1 ) , if x = 3

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2 y = 3 ( 2 x 5 ) , if x = 6

( 6 , 21 2 )

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8 y = 7 ( 8 x + 2 ) , if x = 0

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b = 4 a 12 , if a = 7

( 7 , 40 )

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b = 5 a + 21 , if a = 9

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4 b 6 = 2 a + 1 , if a = 0

( 0 , 7 4 )

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5 m + 11 = n + 1 , if n = 4

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3 ( t + 2 ) = 4 ( s 9 ) , if s = 1

( 1 , 38 3 )

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7 ( t 6 ) = 10 ( 2 s ) , if s = 5

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y = 0 x + 5 , if x = 1

( 1 , 5 )

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2 y = 0 x 11 , if x = 7

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y = 0 x + 10 , if x = 3

( 3 , 10 )

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5 y = 0 x 1 , if x = 0

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y = 0 ( x 1 ) + 6 , if x = 1

( 1 , 6 )

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y = 0 ( 3 x + 9 ) 1 , if x = 12

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Calculator problems

An examination of the winning speeds in the Indianapolis 500 automobile race from 1961 to 1970 produces the equation y = 1.93 x + 137.60 , where x is the number of years from 1960 and y is the winning speed. Statistical methods were used to obtain the equation, and, for a given year, the equation gives only the approximate winning speed. Use the equation y = 1.93 x + 137.60 to find the approximate winning speed in

  1. 1965
  2. 1970
  3. 1986
  4. 1990

(a) Approximately 147 mph using ( 5 , 147.25 )
(b) Approximately 157 mph using ( 10 , 156.9 )
(c) Approximately 188 mph using ( 26 , 187.78 )
(d) Approximately 196 mph using ( 30 , 195.5 )

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In electricity theory, Ohm’s law relates electrical current to voltage by the equation y = 0.00082 x , where x is the voltage in volts and y is the current in amperes. This equation was found by statistical methods and for a given voltage yields only an approximate value for the current. Use the equation y = 0.00082 x to find the approximate current for a voltage of

  1. 6 volts
  2. 10 volts

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Statistical methods have been used to obtain a relationship between the actual and reported number of German submarines sunk each month by the U.S. Navy in World War II. The equation expressing the approximate number of actual sinkings, y , for a given number of reported sinkings, x , is y = 1.04 x + 0.76 . Find the approximate number of actual sinkings of German submarines if the reported number of sinkings is

  1. 4
  2. 9
  3. 10

(a) Approximately 5 sinkings using ( 4 , 4.92 )
(b) Approximately 10 sinkings using ( 9 , 10.12 )
(c) Approximately 11 sinkings using ( 10 , 11.16 )

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Statistical methods have been used to obtain a relationship between the heart weight (in milligrams) and the body weight (in milligrams) of 10-month-old diabetic offspring of crossbred male mice. The equation expressing the approximate body weight for a given heart weight is y = 0.213 x - 4.44 . Find the approximate body weight for a heart weight of

  1. 210 mg
  2. 245 mg

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Statistical methods have been used to produce the equation y = 0.176 x 0.64 . This equation gives the approximate red blood cell count (in millions) of a dog’s blood, y , for a given packed cell volume (in millimeters), x . Find the approximate red blood cell count for a packed cell volume of

  1. 40 mm
  2. 42 mm

(a) Approximately 6.4 using ( 40 , 6.4 )
(b) Approximately 4.752 using ( 42 , 7.752 )

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An industrial machine can run at different speeds. The machine also produces defective items, and the number of defective items it produces appears to be related to the speed at which the machine is running. Statistical methods found that the equation y = 0.73 x 0.86 is able to give the approximate number of defective items, y , for a given machine speed, x . Use this equation to find the approximate number of defective items for a machine speed of

  1. 9
  2. 12

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A computer company has found, using statistical techniques, that there is a relationship between the aptitude test scores of assembly line workers and their productivity. Using data accumulated over a period of time, the equation y = 0.89 x 41.78 was derived. The x represents an aptitude test score and y the approximate corresponding number of items assembled per hour. Estimate the number of items produced by a worker with an aptitude score of

  1. 80
  2. 95

(a) Approximately 29 items using ( 80 , 29.42 )
(b) Approximately 43 items using ( 95 , 42.77 )

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Chemists, making use of statistical techniques, have been able to express the approximate weight of potassium bromide, W , that will dissolve in 100 grams of water at T degrees centigrade. The equation expressing this relationship is W = 0.52 T + 54.2 . Use this equation to predict the potassium bromide weight that will dissolve in 100 grams of water that is heated to a temperature of

  1. 70 degrees centigrade
  2. 95 degrees centigrade

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The marketing department at a large company has been able to express the relationship between the demand for a product and its price by using statistical techniques. The department found, by analyzing studies done in six different market areas, that the equation giving the approximate demand for a product (in thousands of units) for a particular price (in cents) is y = 14.15 x + 257.11 . Find the approximate number of units demanded when the price is

  1. $ 0.12
  2. $ 0.15

(a) Approximately 87 units using ( 12 , 87.31 )
(b) Approximately 45 units using ( 15 , 44.86 )

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The management of a speed-reading program claims that the approximate speed gain (in words per minute), G , is related to the number of weeks spent in its program, W , is given by the equation G = 26.68 W 7.44 . Predict the approximate speed gain for a student who has spent

  1. 3 weeks in the program
  2. 10 weeks in the program

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Exercises for review

( [link] ) Find the product. ( 4 x 1 ) ( 3 x + 5 ) .

12 x 2 + 17 x 5

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( [link] ) Find the product. ( 5 x + 2 ) ( 5 x 2 ) .

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( [link] ) Solve the equation 6 [ 2 ( x 4 ) + 1 ] = 3 [ 2 ( x 7 ) ] .

x = 0

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( [link] ) Solve the inequality 3 a ( a 5 ) a + 10 .

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( [link] ) Solve the compound inequality 1 < 4 y + 11 < 27 .

3 < y < 4

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Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
Bharti
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
Daniel
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
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
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
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.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
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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