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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

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
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Sherica
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what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
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a perfect square v²+2v+_
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Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
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if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
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is it 3×y ?
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J, combine like terms 7x-4y
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what is the problem that i will help you to self with?
Asali
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China
Cied
types of nano material
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I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
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what is nano technology
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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
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
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how did you get the value of 2000N.What calculations are needed to arrive at it
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how do you translate this in Algebraic Expressions
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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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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