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By the end of this section, you will be able to:
  • Plot points on a rectangular coordinate system
  • Identify points on a graph
  • Verify solutions to an equation in two variables
  • Complete a table of solutions to a linear equation
  • Find solutions to linear equations in two variables

Before you get started, take this readiness quiz.

  1. Evaluate: x + 3 when x = −1 .
    If you missed this problem, review Add Integers .
  2. Evaluate: 2 x 5 y when x = 3 , y = −2 .
    If you missed this problem, review Multiply and Divide Integers .
  3. Solve for y : 40 4 y = 20 .
    If you missed this problem, review Solve Equations with Variables and Constants on Both Sides .

Plot points on a rectangular coordinate system

Many maps, such as the Campus Map shown in [link] , use a grid system to identify locations. Do you see the numbers 1 , 2 , 3 , and 4 across the top and bottom of the map and the letters A, B, C, and D along the sides? Every location on the map can be identified by a number and a letter.

For example, the Student Center is in section 2B. It is located in the grid section above the number 2 and next to the letter B. In which grid section is the Stadium? The Stadium is in section 4D.

The figure shows a labeled grid representing the Campus Map. The columns are labeled 1 through 4 and the rows are labeled A through D. At position A-1 is the title Parking Garage. At position A-4 is a rectangle labeled Residence Halls. At position B-2 is a rectangle labeled Student Center. At position B-3 is a rectangle labeled Engineering Building. At position C-1 is a rectangle labeled Taylor Hall. At position C-2 is a rectangle labeled Library.  At position C-4 is a rectangle labeled Tiger Field. At position D-4 is a rectangle labeled Stadium.

Use the map in [link] .

  1. Find the grid section of the Residence Halls.
  2. What is located in grid section 4C?


  1. Read the number below the Residence Halls, 4 , and the letter to the side, A. So the Residence Halls are in grid section 4A.
  2. Find 4 across the bottom of the map and C along the side. Look below the 4 and next to the C. Tiger Field is in grid section 4C.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Use the map in [link] .

  1. Find the grid section of Taylor Hall.
  2. What is located in section 3B?
  1. 1C
  2. Engineering Building
Got questions? Get instant answers now!

Use the map in [link] .

  1. Find the grid section of the Parking Garage.
  2. What is located in section 2C?
  1. 1A
  2. Library
Got questions? Get instant answers now!

Just as maps use a grid system to identify locations, a grid system is used in algebra to show a relationship between two variables in a rectangular coordinate system. To create a rectangular coordinate system, start with a horizontal number line. Show both positive and negative numbers as you did before, using a convenient scale unit. This horizontal number line is called the x -axis    .

The figure shows a number line with integer values labeled from -5 to 5.

Now, make a vertical number line passing through the x -axis at 0 . Put the positive numbers above 0 and the negative numbers below 0 . See [link] . This vertical line is called the y -axis    .

Vertical grid lines pass through the integers marked on the x -axis . Horizontal grid lines pass through the integers marked on the y -axis . The resulting grid is the rectangular coordinate system.

The rectangular coordinate system is also called the x - y plane, the coordinate plane, or the Cartesian coordinate system (since it was developed by a mathematician named René Descartes.)

The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7.  An arrow points to the horizontal axis with the label “x-axis”. An arrow points to the vertical axis with label “y-axis”. An arrow points to the intersection of the axes with label “origin”.
The rectangular coordinate system.

The x -axis and the y -axis form the rectangular coordinate system. These axes divide a plane into four areas, called quadrants    . The quadrants are identified by Roman numerals, beginning on the upper right and proceeding counterclockwise. See [link] .

The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. The top-right portion of the plane is labeled “I”, the top-left portion of the plane is labeled “II”, the bottom-left portion of the plane is labelled “III” and the bottom-right portion of the plane is labeled “IV”
The four quadrants of the rectangular coordinate system

In the rectangular coordinate system, every point is represented by an ordered pair    . The first number in the ordered pair is the x -coordinate of the point, and the second number is the y -coordinate of the point.

Questions & Answers

how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
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The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
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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
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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
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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.
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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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