<< Chapter < Page Chapter >> Page >

Find three solutions to the equation: 2 x + 3 y = 6 .

Answers will vary.

Got questions? Get instant answers now!

Find three solutions to the equation: 4 x + 2 y = 8 .

Answers will vary.

Got questions? Get instant answers now!

Let’s find some solutions to another equation now.

Find three solutions to the equation x 4 y = 8 .

Solution

. . .
Choose a value for x or y . . . .
Substitute it into the equation. . . .
Solve. . . .
Write the ordered pair. ( 0 , −2 ) ( 8 , 0 ) ( 20 , 3 )

So ( 0 , −2 ) , ( 8 , 0 ) , and ( 20 , 3 ) are three solutions to the equation x 4 y = 8 .

x 4 y = 8
x y ( x , y )
0 −2 ( 0 , −2 )
8 0 ( 8 , 0 )
20 3 ( 20 , 3 )

Remember, there are an infinite number of solutions to each linear equation. Any point you find is a solution if it makes the equation true.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find three solutions to the equation: 4 x + y = 8 .

Answers will vary.

Got questions? Get instant answers now!

Find three solutions to the equation: x + 5 y = 10 .

Answers will vary.

Got questions? Get instant answers now!

Key concepts

  • Sign Patterns of the Quadrants
    Quadrant I Quadrant II Quadrant III Quadrant IV
    ( x , y ) ( x , y ) ( x , y ) ( x , y )
    (+,+) (−,+) (−,−) (+,−)
  • Coordinates of Zero
    • Points with a y- coordinate equal to 0 are on the x- axis, and have coordinates ( a , 0).
    • Points with a x- coordinate equal to 0 are on the y- axis, and have coordinates ( 0, b ).
    • The point (0, 0) is called the origin. It is the point where the x- axis and y- axis intersect.

Practice makes perfect

Plot Points on a Rectangular Coordinate System

In the following exercises, plot each point on a coordinate grid.

In the following exercises, plot each point on a coordinate grid and identify the quadrant in which the point is located.

  1. ( −4 , 2 )
  2. ( −1 , −2 )
  3. ( 3 , −5 )
  4. ( 2 , 5 2 )

Got questions? Get instant answers now!

  1. ( −2 , −3 )
  2. ( 3 , −3 )
  3. ( −4 , 1 )
  4. ( 1 , 3 2 )


The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The quadrants are labeled I, II, III, and IV. The point (-1, 1) is labeled a, the point (-2, -1) is labeled b. The point (1, -4) is labeled c, and the point (3, 7/2) is labeled d.

Got questions? Get instant answers now!

  1. ( −1 , 1 )
  2. ( −2 , −1 )
  3. ( 1 , −4 )
  4. ( 3 , 7 2 )

Got questions? Get instant answers now!

  1. ( 3 , -2 )
  2. ( −3 , 2 )
  3. ( -3 , −2 )
  4. ( 3 , 2 )


The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point (3, -2) is labeled a, the point (-3, 2) is labeled b. The point (-3, -2) is labeled c, and the point (3, 2) is labeled d.

Got questions? Get instant answers now!

  1. ( 4 , -1 )
  2. ( −4 , 1 )
  3. ( -4 , -1 )
  4. ( 4 , 1 )

Got questions? Get instant answers now!

  1. ( −2 , 0 )
  2. ( −3 , 0 )
  3. ( 0 , 4 )
  4. ( 0 , 2 )


The graph shows the x y-coordinate plane. The x and y-axis each run from -6 to 6. The point (-2, 0) is labeled a, the point (-3, 0) is labeled b. The point (0, 4) is labeled c, and the point (0, 2) is labeled d.

Got questions? Get instant answers now!

Identify Points on a Graph

In the following exercises, name the ordered pair of each point shown.

Verify Solutions to an Equation in Two Variables

In the following exercises, determine which ordered pairs are solutions to the given equation.

2 x + y = 6

  1. ( 1 , 4 )
  2. ( 3 , 0 )
  3. ( 2 , 3 )

,

Got questions? Get instant answers now!

x + 3 y = 9

  1. ( 0 , 3 )
  2. ( 6 , 1 )
  3. ( -3 , -3 )
Got questions? Get instant answers now!

4 x 2 y = 8

  1. ( 3 , 2 )
  2. ( 1 , 4 )
  3. ( 0 , -4 )

,

Got questions? Get instant answers now!

3 x 2 y = 12

  1. ( 4 , 0 )
  2. ( 2 , -3 )
  3. ( 1 , 6 )

Got questions? Get instant answers now!

y = 4 x + 3

  1. ( 4 , 3 )
  2. ( -1 , -1 )
  3. ( 1 2 , 5 )

,

Got questions? Get instant answers now!

y = 2 x 5

  1. ( 0 , -5 )
  2. ( 2 , 1 )
  3. ( 1 2 , -4 )

Got questions? Get instant answers now!

y = 1 2 x 1

  1. ( 2 , 0 )
  2. ( -6 , -4 )
  3. ( -4 , -1 )

,

Got questions? Get instant answers now!

y = 1 3 x + 1

  1. ( -3 , 0 )
  2. ( 9 , 4 )
  3. ( -6 , -1 )

Got questions? Get instant answers now!

Find Solutions to Linear Equations in Two Variables

In the following exercises, complete the table to find solutions to each linear equation.

y = 2 x 4

x y ( x , y )
−1
0
2
x y ( x , y )
−1 −6 ( −1 , −6 )
0 −4 ( 0 , −4 )
2 0 ( 2 , 0 )
Got questions? Get instant answers now!

y = 3 x 1

x y ( x , y )
−1
0
2
Got questions? Get instant answers now!

y = x + 5

x y ( x , y )
−2
0
3
x y ( x , y )
−2 7 ( −2 , 7 )
0 5 ( 0 , 5 )
3 2 ( 3 , 2 )
Got questions? Get instant answers now!

y = 1 3 x + 1

x y ( x , y )
0
3
6
Got questions? Get instant answers now!

y = 3 2 x 2

x y ( x , y )
−2
0
2
x y ( x , y )
−2 1 ( −2 , 1 )
0 −2 ( 0 , −2 )
2 −5 ( 2 , −5 )
Got questions? Get instant answers now!

x + 2 y = 8

x y ( x , y )
0
4
0
Got questions? Get instant answers now!

Everyday math

Weight of a baby Mackenzie recorded her baby’s weight every two months. The baby’s age, in months, and weight, in pounds, are listed in the table, and shown as an ordered pair in the third column.

Plot the points on a coordinate grid.

Age Weight ( x , y )
0 7 ( 0 , 7 )
2 11 ( 2 , 11 )
4 15 ( 4 , 15 )
6 16 ( 6 , 16 )
8 19 ( 8 , 19 )
10 20 ( 10 , 20 )
12 21 ( 12 , 21 )

Why is only Quadrant I needed?

  1. This figure shows points plot on the x y coordinate plane. There are 7 points graphed without labeled at approximately the points “ordered pair 0, 7”, “ordered pair 2, 11”, “ordered pair 4, 15”, “ordered pair 6, 16”, “ordered pair 8, 19”, “ordered pair 10, 20”, “ordered pair 12, 21”.
  2. Age and weight are only positive.

Got questions? Get instant answers now!

Weight of a child Latresha recorded her son’s height and weight every year. His height, in inches, and weight, in pounds, are listed in the table, and shown as an ordered pair in the third column.

Plot the points on a coordinate grid.

Height x Weight y ( x , y )
28 22 ( 28 , 22 )
31 27 ( 31 , 27 )
33 33 ( 33 , 33 )
37 35 ( 37 , 35 )
40 41 ( 40 , 41 )
42 45 ( 42 , 45 )

Why is only Quadrant I needed?

Got questions? Get instant answers now!

Writing exercises

Have you ever used a map with a rectangular coordinate system? Describe the map and how you used it.

Answers may vary.

Got questions? Get instant answers now!

How do you determine if an ordered pair is a solution to a given equation?

Got questions? Get instant answers now!

Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

.

If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

Questions & Answers

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
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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
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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
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
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
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
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
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
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit 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
Good
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
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
Practice Key Terms 7

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Prealgebra' conversation and receive update notifications?

Ask