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

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

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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.

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

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

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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 )

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  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.

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

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  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.

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

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  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.

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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 )

,

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x + 3 y = 9

  1. ( 0 , 3 )
  2. ( 6 , 1 )
  3. ( -3 , -3 )
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4 x 2 y = 8

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

,

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3 x 2 y = 12

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

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y = 4 x + 3

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

,

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y = 2 x 5

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

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y = 1 2 x 1

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

,

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y = 1 3 x + 1

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

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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 )
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y = 3 x 1

x y ( x , y )
−1
0
2
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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 )
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y = 1 3 x + 1

x y ( x , y )
0
3
6
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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 )
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x + 2 y = 8

x y ( x , y )
0
4
0
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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.

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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?

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Writing exercises

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

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

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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

what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
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

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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