{"id":921,"date":"2021-09-25T22:11:36","date_gmt":"2021-09-25T22:11:36","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&p=921"},"modified":"2024-02-09T20:09:49","modified_gmt":"2024-02-09T20:09:49","slug":"6-4-3-use-the-slope-formula","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/6-4-3-use-the-slope-formula\/","title":{"raw":"6.4.3: The Slope Formula","rendered":"6.4.3: The Slope Formula"},"content":{"raw":"
\r\n
\r\n
\r\n
\r\n
\r\n
\r\n

Learning Outcomes<\/h3>\r\n
    \r\n \t
  • Use the slope formula to find the slope of a line between two points<\/li>\r\n \t
  • Find the slope of horizontal and vertical lines<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n

    The Slope Formula<\/h2>\r\nWe\u2019ve seen that we can find the slope of a line on a graph by measuring the rise and the run. We can also find the slope of a line without its graph if we know the coordinates of any two points on that line. Every point has a set of coordinates: an\u00a0[latex]x[\/latex]-value and a [latex]y[\/latex]-value, written as an ordered pair [latex](x, y)[\/latex]. The [latex]x[\/latex] value tells yus where a point is horizontally. The [latex]y[\/latex] value tells us where the point is vertically.\r\n\r\nConsider two points on a line\u2014Point 1 and Point 2. Point 1 has coordinates [latex]\\left(x_{1},y_{1}\\right)[\/latex]\u00a0and Point 2 has coordinates [latex]\\left(x_{2},y_{2}\\right)[\/latex].\r\n
    \"A<\/div>\r\nThe rise<\/strong><\/em> is the vertical distance between the two points, which is the difference between their \u00a0[latex]y[\/latex]-coordinates. That makes the rise [latex]\\left(y_{2}-y_{1}\\right)[\/latex]. The run<\/strong><\/em> between these two points is the difference in the [latex]x[\/latex]-coordinates, or [latex]\\left(x_{2}-x_{1}\\right)[\/latex].\r\n\r\nSo, [latex] \\displaystyle \\text{Slope}=\\frac{\\text{rise}}{\\text{run}}[\/latex] or [latex] \\displaystyle m=\\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}[\/latex].\r\n\r\nTo see how the rise and run relate to the coordinates of the two points, let\u2019s take a look at the slope of the line between the points [latex]\\left(2,3\\right)[\/latex] and [latex]\\left(7,6\\right)[\/latex].\r\n\r\n\"The\r\n\r\nSince we have two points, we will use subscript notation.\r\n

    [latex]\\stackrel{{x}_{1},{y}_{1}}{\\left(2,3\\right)}\\stackrel{{x}_{2},{y}_{2}}{\\left(7,6\\right)}[\/latex]<\/p>\r\nOn the graph, we can count the rise of [latex]3[\/latex]. The rise can also be found by subtracting the [latex]y\\text{-coordinates}[\/latex] of the points.\r\n

    [latex]\\begin{array}{c}{y}_{2}-{y}_{1}\\\\ 6 - 3\\\\ 3\\end{array}[\/latex]<\/p>\r\nWe can count a run of [latex]5[\/latex] from the graph. The run can also be found by subtracting the [latex]x\\text{-coordinates}[\/latex].\r\n

    [latex]\\begin{array}{c}{x}_{2}-{x}_{1}\\\\ 7 - 2\\\\ 5\\end{array}[\/latex]<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    We know<\/td>\r\n[latex]m={\\frac{\\text{rise}}{\\text{run}}}[\/latex]<\/td>\r\n<\/tr>\r\n
    So<\/td>\r\n[latex]m={\\frac{3}{5}}[\/latex]<\/td>\r\n<\/tr>\r\n
    We rewrite the rise and run by putting in the coordinates.<\/td>\r\n[latex]m={\\frac{6 - 3}{7 - 2}}[\/latex]<\/td>\r\n<\/tr>\r\n
    But [latex]6[\/latex] is the [latex]y[\/latex] -coordinate of the second point, [latex]{y}_{2}[\/latex]and [latex]3[\/latex] is the [latex]y[\/latex] -coordinate of the first point [latex]{y}_{1}[\/latex] .\r\n\r\nSo we can rewrite the rise using subscript notation.<\/td>\r\n[latex]m={\\frac{{y}_{2}-{y}_{1}}{7 - 2}}[\/latex]<\/td>\r\n<\/tr>\r\n
    Also [latex]7[\/latex] is the [latex]x[\/latex] -coordinate of the second point, [latex]{x}_{2}[\/latex]and [latex]2[\/latex] is the [latex]x[\/latex] -coordinate of the first point [latex]{x}_{2}[\/latex] .\r\n\r\nSo we rewrite the run using subscript notation.<\/td>\r\n[latex]m={\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe\u2019ve shown that [latex]m={\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}}[\/latex] is another version of [latex]m={\\frac{\\text{rise}}{\\text{run}}}[\/latex]. We can use this formula to find the slope of a line when we kmow two points on the line.\r\n
    \r\n

    Slope Formula<\/h3>\r\nThe slope of the line between two points [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex] is\r\n

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

    \r\n

    example<\/h3>\r\nFind the slope of the line between the points [latex]\\left(1,2\\right)[\/latex] and [latex]\\left(4,5\\right)[\/latex].\r\n\r\nSolution\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    We\u2019ll call [latex]\\left(1,2\\right)[\/latex] point #1 and [latex]\\left(4,5\\right)[\/latex] point #2.<\/td>\r\n[latex]\\stackrel{{x}_{1},{y}_{1}}{\\left(1,2\\right)}\\text{and}\\stackrel{{x}_{2},{y}_{2}}{\\left(4,5\\right)}[\/latex]<\/td>\r\n<\/tr>\r\n
    Use the slope formula.<\/td>\r\n[latex]m={\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}}[\/latex]<\/td>\r\n<\/tr>\r\n
    Substitute the values in the slope formula:<\/td>\r\n<\/td>\r\n<\/tr>\r\n
    [latex]y[\/latex] of the second point minus [latex]y[\/latex] of the first point<\/td>\r\n[latex]m={\\frac{5 - 2}{{x}_{2}-{x}_{1}}}[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]x[\/latex] of the second point minus [latex]x[\/latex] of the first point<\/td>\r\n[latex]m={\\frac{5 - 2}{4 - 1}}[\/latex]<\/td>\r\n<\/tr>\r\n
    Simplify the numerator and the denominator.<\/td>\r\n[latex]m={\\frac{3}{3}}[\/latex]<\/td>\r\n<\/tr>\r\n
    <\/td>\r\n[latex]m=1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nLet\u2019s confirm this by counting out the slope on the graph.\r\n\r\n\"The\r\n\r\nThe rise is [latex]3[\/latex] and the run is [latex]3[\/latex], so\r\n\r\n[latex]\\begin{array}{}\\\\ m=\\frac{\\text{rise}}{\\text{run}}\\hfill \\\\ m={\\Large\\frac{3}{3}}\\hfill \\\\ m=1\\hfill \\end{array}[\/latex]\r\n\r\n<\/div>\r\nIt is important to remember that the slope is the same no matter which order we select the points.\u00a0 Previously, whenever we found the slope by looking at the graph, we always selected our points from left to right so that our run was always a positive value.\u00a0 Now, let\u2019s take a look at an example in which we select our points from right to left.\r\n\r\nThe point [latex](0,2)[\/latex] is indicated as Point 1, and [latex](\u22122,6)[\/latex] as Point 2. So you are going to move from Point 1 to Point 2. A triangle is drawn in above the line to help illustrate the rise and run.\r\n
    \"A<\/div>\r\nWe can see from the graph that the rise going from Point 1 to Point 2 is [latex]4[\/latex], because we are moving [latex]4[\/latex] units in a positive direction (up). The run is [latex]\u22122[\/latex], because we are moving in a negative direction (left) [latex]2[\/latex] units. Using the slope formula,\r\n

    [latex] \\displaystyle \\text{Slope}=\\frac{\\text{rise}}{\\text{run}}=\\frac{4}{-2}=-2[\/latex].<\/p>\r\nWe do not need the graph to find the slope. We can just use the coordinates, keeping careful track of which is Point 1 and which is Point 2. Let\u2019s organize the information about the two points:\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    Name<\/th>\r\nOrdered Pair<\/th>\r\nCoordinates<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
    Point 1<\/td>\r\n[latex](0,2)[\/latex]<\/td>\r\n[latex]\\begin{array}{l}x_{1}=0\\\\y_{1}=2\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n
    Point 2<\/td>\r\n[latex](\u22122,6)[\/latex]<\/td>\r\n[latex]\\begin{array}{l}x_{2}=-2\\\\y_{2}=6\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe slope, [latex]m=\\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\\frac{6-2}{-2-0}=\\frac{4}{-2}=-2[\/latex]. The slope of the line,\u00a0m<\/i>, is [latex]\u22122[\/latex].\r\n\r\nRemember, it doesn\u2019t matter which point is designated as Point 1 and which is Point 2. We could have called [latex](\u22122,6)[\/latex] Point 1, and [latex](0,2)[\/latex] Point 2. In that\u00a0case, putting the coordinates into the slope formula produces the equation [latex]m=\\frac{2-6}{0-\\left(-2\\right)}=\\frac{-4}{2}=-2[\/latex]. Once again, the slope is [latex]m=-2[\/latex]. That\u2019s the same slope as before. The important thing is to be consistent when we subtract: we must always subtract in the same order [latex]\\left(y_{2}-y_{1}\\right)[\/latex]\u00a0and [latex]\\left(x_{2}-x_{1}\\right)[\/latex].\r\n
    \r\n

    try it<\/h3>\r\n