{"id":6639,"date":"2020-10-03T15:22:30","date_gmt":"2020-10-03T15:22:30","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/slcc-beginalgebra\/?post_type=chapter&p=6639"},"modified":"2026-02-05T07:33:19","modified_gmt":"2026-02-05T07:33:19","slug":"3-1-rectangular-coordinate-system-and-ordered-pairs","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-elementaryalgebra\/chapter\/3-1-rectangular-coordinate-system-and-ordered-pairs\/","title":{"raw":"3.1: Cartesian Coordinate System and Ordered Pairs","rendered":"3.1: Cartesian Coordinate System and Ordered Pairs"},"content":{"raw":"
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Section 3.1 Learning Objectives<\/h3>\r\n3.1: Cartesian Coordinate Plane and Ordered Pairs<\/strong>\r\n
    \r\n \t
  • Given a point in the coordinate plane, determine the ordered pair that describes the point<\/li>\r\n \t
  • Plot points in the coordinate plane<\/li>\r\n \t
  • Given an ordered pair, identify in which quadrant of the coordinate plane it is located<\/li>\r\n<\/ul>\r\n<\/div>\r\n \r\n\r\nThe coordinate plane<\/b> was developed centuries ago (in 1637, to be exact) and refined by the French mathematician Ren\u00e9 Descartes. In his honor, the system is sometimes called the Cartesian coordinate system. In this chapter, we will study how the coordinate plane can be used to plot points and graph lines. This system allows us to describe algebraic relationships in a visual sense, and also helps us create and interpret algebraic concepts.\r\n

    The components of the coordinate plane<\/h2>\r\nYou have likely used a coordinate plane before. For example, have you ever used a gridded overlay to map the position of an object? (This is often done with road maps, too.)\r\n\r\n\"A\r\n\r\nThis \u201cmap\u201d uses a horizontal and vertical grid to convey information about an object\u2019s location. Notice that the letters A\u2013F are listed along the top, and the numbers 1\u20136 are listed along the left edge. The general location of any item on this map can be found by using the letter and number of its grid square. For example, you can find the item that exists at square \u201c4F\u201d by moving your finger along the horizontal to letter F and then straight down so you are in line with the 4. You\u2019ll find a blue disc is at this location on the map.\r\n\r\nThe coordinate plane has similar elements to the grid shown above. It consists of a horizontal axis<\/b> and a vertical axis, number lines that intersect at right angles (perpendicular to each other),\r\n\r\n\"A\r\n\r\nTypically, the horizontal axis in the coordinate plane is called the x-axis<\/b>\u00a0and the vertical axis is called the y-axis<\/b>. The point at which the two axes intersect is called the origin<\/b>. The origin is at 0 on the x-<\/i>axis and 0 on the y-<\/i>axis.\r\n\r\nLocations on the coordinate plane are described as ordered pairs<\/b>. An ordered pair tells you the location of a point by relating the point\u2019s location along the x-<\/i>axis (the first value of the ordered pair) and along the y<\/i>-axis (the second value of the ordered pair).\r\n\r\nIn an ordered pair, written (x<\/i>, y<\/i>), the first value is called the x-coordinate<\/b> and the second value is the y-coordinate<\/b>.\u00a0Since the origin has an x-<\/i>coordinate of 0 and a y-<\/i>coordinate of 0, its ordered pair is written (0, 0).\r\n

    Describe the point shown as an ordered pair<\/h2>\r\nConsider the point below.\r\n\r\n\"Grid\r\n\r\nTo identify the location of this point, start at the origin (0, 0) and move right along the x-<\/i>axis until you are under the point. Look at the label on the x-<\/i>axis. The 4 indicates that, from the origin, you have traveled four units to the right along the x<\/i>-axis. This is the x-<\/i>coordinate, the first number in the ordered pair.\r\n\r\nFrom 4 on the x-<\/i>axis move up to the point and notice the number with which it aligns on the y-<\/i>axis. The 3 indicates that, after leaving the x<\/i>-axis, you traveled 3 units up in the vertical direction, the direction of the y<\/i>-axis. This number is the y-<\/i>coordinate, the second number in the ordered pair. With an x-<\/i>coordinate of 4 and a y-<\/i>coordinate of 3, you have the ordered pair (4, 3).\r\n\r\nLet\u2019s look at another example.\r\n
    \r\n

    Example 1<\/h3>\r\nDescribe the point shown as an ordered pair.\r\n\r\n\"A\r\n\r\n[reveal-answer q=\"668288\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"668288\"]\r\n\r\nBegin at the origin and move along the x-<\/i>axis until you are directly below the point. This is the x-<\/i>coordinate and is written first in the ordered pair.\r\n

    (5, y<\/i>)<\/p>\r\nMove from 5 up to the ordered pair and read the number on the y-<\/i>axis. This is the y-<\/i>coordinate and is written second in the ordered pair.\r\n

    (5, 2)<\/p>\r\n\r\n

    Answer<\/h4>\r\nThe point shown as an ordered pair is (5, 2).[\/hidden-answer]\r\n\r\n<\/div>\r\n

    <\/h3>\r\nhttps:\/\/youtu.be\/c9WVU34MY5Q\r\n

    Plotting points in the coordinate plane<\/h2>\r\nNow that you know how to use the x-<\/i> and y-<\/i>axes, you can plot an ordered pair as well. Just remember, both processes start at the origin\u2014the beginning! The example that follows shows how to graph the ordered pair (1,3).\r\n
    \r\n

    Example 2<\/h3>\r\nPlot the point (1, 3).\r\n\r\n[reveal-answer q=\"28562\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"28562\"]\r\n\r\nThe x-<\/i>coordinate is 1 because it comes first in the ordered pair. Start at the origin and move a distance of 1 unit in the positive x<\/em>-direction (to the right) from the origin along the x-<\/i>axis.\r\n\r\nThe y-<\/i>coordinate is 3 because it comes second in the ordered pair. From here move directly 3 units in the positive y<\/em>-direction (up). If you look over to the y-<\/i>axis, you should be lined up with 3 on that axis.\r\n

    Answer<\/h4>\r\nDraw a point at this location and label the point (1, 3).\"1[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the previous example, both the x-<\/i> and y-<\/i>coordinates were positive. When one (or both) of the coordinates of an ordered pair is negative, you will need to move in the negative direction along one or both axes. Consider the example below in which both coordinates are negative.\r\n
    \r\n

    Example 3<\/h3>\r\nPlot the point [latex](\u22124,\u22122)[\/latex].\r\n\r\n\"Coordinate\r\n\r\nThe x-<\/i>coordinate is [latex]\u22124[\/latex] because it comes first in the ordered pair. Start at the origin and move 4 units in the negative direction (left) along the x-<\/i>axis.\r\n\r\nThe y-<\/i>coordinate is [latex]\u22122[\/latex] because it comes second in the ordered pair. Now move 2 units in the negative y<\/em>-direction (down). If you look over to the y-<\/i>axis, you should be lined up with [latex]\u22122[\/latex] on that axis.\r\n\r\n[reveal-answer q=\"118522\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"118522\"]Draw a point at this location and label the point [latex](\u22124,\u22122)[\/latex].[\/hidden-answer]\r\n\r\n<\/div>\r\nThe steps for plotting a point are summarized below.\r\n
    \r\n

    Steps for Plotting an Ordered Pair (x<\/i>, y<\/i>) in the Coordinate Plane<\/h3>\r\n
      \r\n \t
    • Determine the x-<\/i>coordinate. Beginning at the origin, move horizontally, the direction of the x<\/i>-axis, the distance indicated by the x-<\/i>coordinate. If the x-<\/i>coordinate is positive, move to the right; if the x-<\/i>coordinate is negative, move to the left.<\/li>\r\n \t
    • Determine the y-<\/i>coordinate. Beginning at the x-<\/i>coordinate, move vertically, the direction of the y<\/i>-axis, the distance indicated by the y-<\/i>coordinate. If the y-<\/i>coordinate is positive, move up; if the y-<\/i>coordinate is negative, move down.<\/li>\r\n \t
    • Draw a point at the ending location. Label the point with the ordered pair.<\/li>\r\n<\/ul>\r\n<\/div>\r\nhttps:\/\/youtu.be\/p_MESleS3mw\r\n\r\nThe next example include fractional coordinates.\r\n
      \r\n

      Example 4<\/h3>\r\nPlot the following points on the coordinate plane.\r\n\r\nA<\/strong> [latex](\\frac{1}{2}, 8) [\/latex], B<\/strong>\u00a0[latex](-7, \\frac{-1}{4}) [\/latex], C<\/strong>\u00a0[latex](\\frac{5}{2}, 2) [\/latex]\r\n\r\n[reveal-answer q=\"917341\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"917341\"]\r\n\r\nThe ordered pairs are plotted and labeled below. Notice the scale on the graph and that each grid mark line represents\u00a0[latex]\\frac{1}{4}[\/latex]th of a unit. While we would not always want to use this scaling necessarily, we can always scale to make the particular given fractions easier to locate.\r\n\r\n\"3\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nSometimes the scales on the x-axis and y-axis won't match. It's always important to pay attention to the scales so that you plot points correctly, as seen in the next example:\r\n
      \r\n

      Example 5<\/h3>\r\nPlot the following points on a coordinate plane.\r\n\r\nA\u00a0<\/strong>\u00a0[latex](100, 2) [\/latex],\u00a0B\u00a0<\/strong>[latex](-40, 8) [\/latex],\u00a0C\u00a0<\/strong>[latex](30, -4) [\/latex]\r\n\r\n[reveal-answer q=\"547704\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"547704\"]\r\n\r\nTo fit all our points on a manageable graph, we will make our x-axis in units of 20, while the y-axis remains in units of 1. It is always important to look for differences in units between the x and y-axis, as they will not always match.\r\n\r\n\"3\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n \r\n

      Determine the quadrant an ordered pair is located in<\/h2>\r\nThe intersecting x-<\/i> and y-<\/i>axes of the coordinate plane divide it\u00a0into four distinct sections. These four sections are called quadrants<\/b>. Quadrants are named using the Roman numerals I, II, III, and IV, beginning with the top right quadrant and moving counterclockwise.\r\n\r\nOrdered pairs within any particular quadrant share certain characteristics. Look at each quadrant in the graph below. What do you notice about the signs of the x-<\/i> and y-<\/i>coordinates of the points within each quadrant?\r\n\r\n\"A\r\n\r\nWithin each quadrant, the signs of the x-<\/i>coordinates and y-<\/i>coordinates of each ordered pair are the same. They also follow a pattern, which is outlined in the table below.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      Quadrant<\/th>\r\nGeneral Form of Point in this Quadrant<\/th>\r\nExample<\/th>\r\nDescription<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
      I<\/td>\r\n[latex](+,+)[\/latex]<\/td>\r\n[latex](5,4)[\/latex]<\/td>\r\nStarting from the origin, go along the x-<\/i>axis in the positive direction (right) and along the y-<\/i>axis in the positive direction (up).<\/td>\r\n<\/tr>\r\n
      II<\/td>\r\n[latex](\u2212,+)[\/latex]<\/td>\r\n[latex](\u22125,4)[\/latex]<\/td>\r\nStarting from the origin, go along the x-<\/i>axis in the negative direction (left) and along the y-<\/i>axis in the positive direction (up).<\/td>\r\n<\/tr>\r\n
      III<\/td>\r\n[latex](\u2212,\u2212)[\/latex]<\/td>\r\n[latex](\u22125,\u22124)[\/latex]<\/td>\r\nStarting from the origin, go along the x-<\/i>axis in the negative direction (left) and along the y-<\/i>axis in the negative direction (down).<\/td>\r\n<\/tr>\r\n
      IV<\/td>\r\n[latex](+,\u2212)[\/latex]<\/td>\r\n[latex](5,\u22124)[\/latex]<\/td>\r\nStarting from the origin, go along the x-<\/i>axis in the positive direction (right) and along the y-<\/i>axis in the negative direction (down).<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nOnce you know about the quadrants in the coordinate plane, you can determine the quadrant of an ordered pair without even graphing it by looking at the chart above. Here is a more visual way to think about it.\r\n\r\n\"Coordinate\r\n\r\nThe example below details how to determine the quadrant location of a point just by thinking about the signs of its coordinates. Thinking about the quadrant location before plotting a point can help you prevent a mistake. It is also useful knowledge for checking that you have plotted a point correctly.\r\n
      \r\n

      Example 6<\/h3>\r\nIn which quadrant is the point [latex](\u22127,10)[\/latex] located?\r\n\r\n[reveal-answer q=\"222353\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"222353\"]\r\n\r\nLook at the signs of the x-<\/i> and y-<\/i>coordinates. For this ordered pair, the signs are [latex](\u2212,+)[\/latex].\r\n

      [latex](\u22127,10)[\/latex]<\/p>\r\nUsing the table or grid above, locate the pattern [latex](\u2212,+)[\/latex].\r\n\r\nPoints with the pattern [latex](\u2212,+)[\/latex] are in Quadrant II.\r\n

      Answer<\/h4>\r\nThe point [latex](\u22127,10)[\/latex] is in Quadrant II.[\/hidden-answer]\r\n\r\n<\/div>\r\n
      \r\n

      Example 7<\/h3>\r\nIn which quadrant is the point [latex](\u221210,\u22125)[\/latex] located?\r\n\r\n[reveal-answer q=\"999799\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"999799\"]\r\n\r\nLook at the signs of the x-<\/i> and y-<\/i>coordinates. For this ordered pair, the signs are [latex](\u2212,\u2212)[\/latex].\r\n

      [latex](\u221210,\u22125)[\/latex]<\/p>\r\nPoints with the pattern [latex](\u2212,\u2212)[\/latex] are in Quadrant III.\r\n\r\nUsing the table or grid above, locate the pattern [latex](\u2212,\u2212)[\/latex].\r\n

      Answer<\/h4>\r\nThe point [latex](\u221210,\u22125)[\/latex] is in Quadrant III.[\/hidden-answer]\r\n\r\n<\/div>\r\nWhat happens if an ordered pair has an x<\/em>- or y-<\/i>coordinate of zero? The example below shows the graph of the ordered pair [latex](0,4)[\/latex].\r\n\r\n\"A\r\n\r\nA point located on one of the axes is not considered to be in a quadrant. It is simply on one of the axes. Whenever the x-<\/i>coordinate is 0, the point is located on the y-<\/i>axis. Similarly, any point that has a y-<\/i>coordinate of 0 will be located on the x-<\/i>axis.\r\n\r\nhttps:\/\/youtu.be\/iTsJsPgcE4E\r\n

      Summary<\/h2>\r\nThe coordinate plane is a system that can be used for graphing and describing points and lines. The coordinate plane is comprised of a horizontal (x<\/i>-) axis and a vertical (y-<\/i>) axis. The intersection of these number lines creates the origin, which is the point [latex](0,0)[\/latex]. The coordinate plane is split into four quadrants. Together, these features of the coordinate system allow for the graphical representation and communication about points, lines, and other algebraic objects.","rendered":"
      \n

      Section 3.1 Learning Objectives<\/h3>\n

      3.1: Cartesian Coordinate Plane and Ordered Pairs<\/strong><\/p>\n

        \n
      • Given a point in the coordinate plane, determine the ordered pair that describes the point<\/li>\n
      • Plot points in the coordinate plane<\/li>\n
      • Given an ordered pair, identify in which quadrant of the coordinate plane it is located<\/li>\n<\/ul>\n<\/div>\n

         <\/p>\n

        The coordinate plane<\/b> was developed centuries ago (in 1637, to be exact) and refined by the French mathematician Ren\u00e9 Descartes. In his honor, the system is sometimes called the Cartesian coordinate system. In this chapter, we will study how the coordinate plane can be used to plot points and graph lines. This system allows us to describe algebraic relationships in a visual sense, and also helps us create and interpret algebraic concepts.<\/p>\n

        The components of the coordinate plane<\/h2>\n

        You have likely used a coordinate plane before. For example, have you ever used a gridded overlay to map the position of an object? (This is often done with road maps, too.)<\/p>\n

        \"A<\/p>\n

        This \u201cmap\u201d uses a horizontal and vertical grid to convey information about an object\u2019s location. Notice that the letters A\u2013F are listed along the top, and the numbers 1\u20136 are listed along the left edge. The general location of any item on this map can be found by using the letter and number of its grid square. For example, you can find the item that exists at square \u201c4F\u201d by moving your finger along the horizontal to letter F and then straight down so you are in line with the 4. You\u2019ll find a blue disc is at this location on the map.<\/p>\n

        The coordinate plane has similar elements to the grid shown above. It consists of a horizontal axis<\/b> and a vertical axis, number lines that intersect at right angles (perpendicular to each other),<\/p>\n

        \"A<\/p>\n

        Typically, the horizontal axis in the coordinate plane is called the x-axis<\/b>\u00a0and the vertical axis is called the y-axis<\/b>. The point at which the two axes intersect is called the origin<\/b>. The origin is at 0 on the x-<\/i>axis and 0 on the y-<\/i>axis.<\/p>\n

        Locations on the coordinate plane are described as ordered pairs<\/b>. An ordered pair tells you the location of a point by relating the point\u2019s location along the x-<\/i>axis (the first value of the ordered pair) and along the y<\/i>-axis (the second value of the ordered pair).<\/p>\n

        In an ordered pair, written (x<\/i>, y<\/i>), the first value is called the x-coordinate<\/b> and the second value is the y-coordinate<\/b>.\u00a0Since the origin has an x-<\/i>coordinate of 0 and a y-<\/i>coordinate of 0, its ordered pair is written (0, 0).<\/p>\n

        Describe the point shown as an ordered pair<\/h2>\n

        Consider the point below.<\/p>\n

        \"Grid<\/p>\n

        To identify the location of this point, start at the origin (0, 0) and move right along the x-<\/i>axis until you are under the point. Look at the label on the x-<\/i>axis. The 4 indicates that, from the origin, you have traveled four units to the right along the x<\/i>-axis. This is the x-<\/i>coordinate, the first number in the ordered pair.<\/p>\n

        From 4 on the x-<\/i>axis move up to the point and notice the number with which it aligns on the y-<\/i>axis. The 3 indicates that, after leaving the x<\/i>-axis, you traveled 3 units up in the vertical direction, the direction of the y<\/i>-axis. This number is the y-<\/i>coordinate, the second number in the ordered pair. With an x-<\/i>coordinate of 4 and a y-<\/i>coordinate of 3, you have the ordered pair (4, 3).<\/p>\n

        Let\u2019s look at another example.<\/p>\n

        \n

        Example 1<\/h3>\n

        Describe the point shown as an ordered pair.<\/p>\n

        \"A<\/p>\n

        Show Solution<\/span><\/p>\n
        \n

        Begin at the origin and move along the x-<\/i>axis until you are directly below the point. This is the x-<\/i>coordinate and is written first in the ordered pair.<\/p>\n

        (5, y<\/i>)<\/p>\n

        Move from 5 up to the ordered pair and read the number on the y-<\/i>axis. This is the y-<\/i>coordinate and is written second in the ordered pair.<\/p>\n

        (5, 2)<\/p>\n

        Answer<\/h4>\n

        The point shown as an ordered pair is (5, 2).<\/p><\/div>\n<\/div>\n<\/div>\n

        <\/h3>\n