{"id":999,"date":"2024-05-01T21:11:23","date_gmt":"2024-05-01T21:11:23","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=999"},"modified":"2025-08-13T15:12:20","modified_gmt":"2025-08-13T15:12:20","slug":"graphing-and-analyzing-linear-equations-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/graphing-and-analyzing-linear-equations-learn-it-4\/","title":{"raw":"Graphing and Analyzing Linear Equations: Learn It 4","rendered":"Graphing and Analyzing Linear Equations: Learn It 4"},"content":{"raw":"

Slope of a Linear Equation<\/strong><\/h2>\r\nThe slope<\/strong> of a line refers to the ratio of the vertical change in y<\/em> over the horizontal change in x<\/em> between any two points on a line. It indicates the direction in which a line slants as well as its steepness. Slope is sometimes described as rise over run.\r\n
[latex]m=\\dfrac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex]<\/div>\r\nIf the slope is positive, the line slants upward to the right. If the slope is negative, the line slants downward to the right. As the slope increases, the line becomes steeper. Some examples are shown below. The lines indicate the following slopes: [latex]m=-3[\/latex], [latex]m=2[\/latex], and [latex]m=\\frac{1}{3}[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Coordinate Coordinate plane with three linear functions plotted[\/caption]\r\n\r\n
\r\n

Slope of a Line<\/h3>\r\nThe slope<\/strong> of a line is a measure of its steepness or the angle at which it tilts, expressed as the ratio of the rise (the vertical change) to the run (the horizontal change) between any two points on the line. It quantifies how much the line goes up or down as it moves from left to right.\r\n
    \r\n \t
  • A positive<\/strong> slope means that the line rises from left to right.<\/li>\r\n \t
  • A negative<\/strong> slope means that the line falls from left to right.<\/li>\r\n \t
  • A slope of\u00a0zero<\/strong> means the line is flat.<\/li>\r\n<\/ul>\r\nMathematically, the slope is calculated by its rise-to-run ratio:\r\n

    [latex]\\text{slope} = \\dfrac{\\text{rise}}{\\text{run}}[\/latex]<\/p>\r\n \r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"301\"]\"A A graph illustrating a straight diagonal line with labels[\/caption]\r\n\r\n \r\n\r\nGiven two points, [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex], the following formula determines the slope of a line containing these points:\r\n

    [latex]m=\\dfrac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex]<\/div>\r\n<\/section>
    Find the slope of the line:\r\n\r\n[caption id=\"attachment_996\" align=\"aligncenter\" width=\"300\"]\"\" A graph with a linear function with labels for rise and run[\/caption]\r\n\r\n[reveal-answer q=\"746921\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"746921\"]\r\n
      \r\n \t
    • Start from a point on the left side of line, such as [latex](2, 1)[\/latex] and move vertically until in line with another point on the line, such as [latex](6, 3)[\/latex]. The rise is [latex]2[\/latex] units. It is positive because you go up. If it is going down, use a negative to represent that.<\/li>\r\n \t
    • Next, move horizontally to the point [latex](6, 3)[\/latex], a point that is to the right of the first point you picked. Count the number of units. The run is [latex]4[\/latex] units. Always make sure that you count to the right.<\/li>\r\n<\/ul>\r\nThen, solve using the formula:\r\n

      [latex]\\text{slope} = \\dfrac{\\text{rise}}{\\text{run}} = \\dfrac{2}{4} = \\dfrac{1}{2}[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

      When you need to calculate the slope of a line, remember that you can select any two points<\/strong> along the line to perform your calculation.\r\n
        \r\n \t
      • Choose points that are clear and easy to identify on the graph. This often means selecting points where the line crosses grid lines, as the coordinates are straightforward to read.<\/li>\r\n \t
      • Start by selecting the clearest leftmost point where the line intersects the grid, then choose a second point to the right, ensuring it also clearly falls on a grid intersection.<\/li>\r\n \t
      • Measure the vertical change (rise), noting if it increases (positive) or decreases (negative), then measure the horizontal distance (run), which is always positive as you move from left to right, to calculate the slope.<\/li>\r\n<\/ul>\r\n<\/section>
        Find the slope of a line that passes through the points [latex]\\left(2,-1\\right)[\/latex] and [latex]\\left(-5,3\\right)[\/latex].\r\n[reveal-answer q=\"688301\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"688301\"]We substitute the y-<\/em>values and the x-<\/em>values into the formula.\r\n
        [latex]\\begin{array}{l}m\\hfill&=\\frac{3-\\left(-1\\right)}{-5 - 2}\\hfill \\\\ \\hfill&=\\frac{4}{-7}\\hfill \\\\ \\hfill&=-\\frac{4}{7}\\hfill \\end{array}[\/latex]<\/div>\r\nThe slope is [latex]-\\frac{4}{7}[\/latex].\r\n\r\nAnalysis of the Solution<\/strong>\r\n\r\nIt does not matter which point is called [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] or [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex]. As long as we are consistent with the order of the y<\/em> terms and the order of the x<\/em> terms in the numerator and denominator, the calculation will yield the same result.\r\n[\/hidden-answer]\r\n\r\n<\/section>
        [ohm2_question hide_question_numbers=1]18920[\/ohm2_question]<\/section>","rendered":"

        Slope of a Linear Equation<\/strong><\/h2>\n

        The slope<\/strong> of a line refers to the ratio of the vertical change in y<\/em> over the horizontal change in x<\/em> between any two points on a line. It indicates the direction in which a line slants as well as its steepness. Slope is sometimes described as rise over run.<\/p>\n

        [latex]m=\\dfrac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex]<\/div>\n

        If the slope is positive, the line slants upward to the right. If the slope is negative, the line slants downward to the right. As the slope increases, the line becomes steeper. Some examples are shown below. The lines indicate the following slopes: [latex]m=-3[\/latex], [latex]m=2[\/latex], and [latex]m=\\frac{1}{3}[\/latex].<\/p>\n

        \"Coordinate
        Coordinate plane with three linear functions plotted<\/figcaption><\/figure>\n
        \n

        Slope of a Line<\/h3>\n

        The slope<\/strong> of a line is a measure of its steepness or the angle at which it tilts, expressed as the ratio of the rise (the vertical change) to the run (the horizontal change) between any two points on the line. It quantifies how much the line goes up or down as it moves from left to right.<\/p>\n

          \n
        • A positive<\/strong> slope means that the line rises from left to right.<\/li>\n
        • A negative<\/strong> slope means that the line falls from left to right.<\/li>\n
        • A slope of\u00a0zero<\/strong> means the line is flat.<\/li>\n<\/ul>\n

          Mathematically, the slope is calculated by its rise-to-run ratio:<\/p>\n

          [latex]\\text{slope} = \\dfrac{\\text{rise}}{\\text{run}}[\/latex]<\/p>\n

           <\/p>\n

          \"A
          A graph illustrating a straight diagonal line with labels<\/figcaption><\/figure>\n

           <\/p>\n

          Given two points, [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex], the following formula determines the slope of a line containing these points:<\/p>\n

          [latex]m=\\dfrac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex]<\/div>\n<\/section>\n
          Find the slope of the line:<\/p>\n
          \"\"
          A graph with a linear function with labels for rise and run<\/figcaption><\/figure>\n