<< Chapter < Page Chapter >> Page >
By the end of this section, you will be able to:
  • Plot points in a rectangular coordinate system
  • Verify solutions to an equation in two variables
  • Complete a table of solutions to a linear equation
  • Find solutions to a linear equation in two variables

Before you get started, take this readiness quiz.

  1. Evaluate x + 3 when x = −1 .
    If you missed this problem, review [link] .
  2. Evaluate 2 x 5 y when x = 3 and y = −2 .
    If you missed this problem, review [link] .
  3. Solve for y : 40 4 y = 20 .
    If you missed this problem, review [link] .

Plot points on a rectangular coordinate system

Just like 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    . The rectangular coordinate system is also called the xy -plane or the ‘coordinate plane’.

The horizontal number line is called the x-axis . The vertical number line is called the y-axis. The x -axis and the y -axis together form the rectangular coordinate system. These axes divide a plane into four regions, 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-axes each run from negative 7 to 7. The top-right portion of the plane is labeled
‘Quadrant’ has the root ‘quad,’ which means ‘four.’

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.

Ordered pair

An ordered pair    , ( x , y ) , gives the coordinates of a point in a rectangular coordinate system.

The ordered pair x y is labeled with the first coordinate x labeled as

The first number is the x -coordinate.

The second number is the y -coordinate.

The phrase ‘ordered pair’ means the order is important. What is the ordered pair of the point where the axes cross? At that point both coordinates are zero, so its ordered pair is ( 0 , 0 ) . The point ( 0 , 0 ) has a special name. It is called the origin    .

The origin

The point ( 0 , 0 ) is called the origin    . It is the point where the x -axis and y -axis intersect.

We use the coordinates to locate a point on the xy -plane. Let’s plot the point ( 1 , 3 ) as an example. First, locate 1 on the x -axis and lightly sketch a vertical line through x = 1 . Then, locate 3 on the y -axis and sketch a horizontal line through y = 3 . Now, find the point where these two lines meet—that is the point with coordinates ( 1 , 3 ) .

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. An arrow starts at the origin and extends right to the number 2 on the x-axis. The point (1, 3) is plotted and labeled. Two dotted lines, one parallel to the x-axis, the other parallel to the y-axis, meet perpendicularly at 1, 3. The dotted line parallel to the x-axis intercepts the y-axis at 3. The dotted line parallel to the y-axis intercepts the x-axis at 1.

Notice that the vertical line through x = 1 and the horizontal line through y = 3 are not part of the graph. We just used them to help us locate the point ( 1 , 3 ) .

Plot each point in the rectangular coordinate system and identify the quadrant    in which the point is located:

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

Solution

The first number of the coordinate pair is the x -coordinate, and the second number is the y -coordinate.

  1. Since x = −5 , the point is to the left of the y -axis. Also, since y = 4 , the point is above the x -axis. The point ( −5 , 4 ) is in Quadrant II.
  2. Since x = −3 , the point is to the left of the y -axis. Also, since y = −4 , the point is below the x -axis. The point ( −3 , −4 ) is in Quadrant III.
  3. Since x = 2 , the point is to the right of the y -axis. Since y = −3 , the point is below the x -axis. The point ( 2 , −3 ) is in Quadrant lV.
  4. Since x = −2 , the point is to the left of the y -axis. Since y = 3 , the point is above the x -axis. The point ( −2 , 3 ) is in Quadrant II.
  5. Since x = 3 , the point is to the right of the y -axis. Since y = 5 2 , the point is above the x -axis. (It may be helpful to write 5 2 as a mixed number or decimal.) The point ( 3 , 5 2 ) is in Quadrant I.
    The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The points (negative 5, 4), (negative 2, 3), (negative 3, negative 4), (3, five halves), and (2, negative 3) are plotted and labeled.
Got questions? Get instant answers now!
Got questions? Get instant answers now!
Practice Key Terms 7

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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

Ask