{"id":931,"date":"2021-09-26T02:58:23","date_gmt":"2021-09-26T02:58:23","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&p=931"},"modified":"2021-12-10T22:53:58","modified_gmt":"2021-12-10T22:53:58","slug":"6-5-2-construct-an-equation-given-one-or-two-points","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/6-5-2-construct-an-equation-given-one-or-two-points\/","title":{"raw":"6.5.2: Constructing an Equation Given Points on a Line","rendered":"6.5.2: Constructing an Equation Given Points on a Line"},"content":{"raw":"
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Learning Outcomes<\/h3>\r\n
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  • Write the equation of a line using slope and a point on the line<\/li>\r\n \t
  • Write the equation of a line using two points on the line<\/li>\r\n<\/ul>\r\n<\/div>\r\n

    Finding the Equation of a Line<\/h2>\r\n

    Given the Slope and a Point on the Line<\/h3>\r\nUsing the slope-intercept equation of a line is straight forward when we know both the slope, [latex]m[\/latex], and the [latex]y[\/latex]-intercept, [latex](0, b)[\/latex], but what if we know the slope and a point on the line that is not the [latex]y[\/latex]-intercept?\r\n\r\nFor example, suppose we know that a line has a slope of [latex]5[\/latex] and that the point [latex](2, 1)[\/latex] lies on the line. To lie on the line\u00a0[latex](2, 1)[\/latex] must satisfy the linear equation [latex]y=5x+b[\/latex]. Remember we know that [latex]m=5[\/latex]. In other words, the point [latex](2, 1)[\/latex] must be a solution of the equation [latex]y=5x+b[\/latex], since it lies on the line. If we replace [latex]x[\/latex] and [latex]y[\/latex] with [latex]2[\/latex] and\u00a0[latex]1[\/latex], respectively, we can solve the resulting equation for\u00a0[latex]b[\/latex].\r\n\r\n[latex]\\begin{equation}\\begin{aligned}y& = mx+b\\;\\;\\;\\;\\;\\;m=5 \\text{ and } (2,1)\\text{ is an }(x, y)\\text{ solution} \\\\ 1& = 5(2)+b \\\\1 & = 10 + b \\\\ -9 & = b\\end{aligned}\\end{equation}[\/latex]\r\n\r\nSo, since\u00a0[latex]m=5[\/latex] and\u00a0[latex]b=-9[\/latex], the equation of the line is\u00a0[latex]y=5x-9[\/latex].\r\n
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    Example<\/h3>\r\nDetermine the equation of the line that has a slope of [latex]3[\/latex] and contains the point [latex](1,4)[\/latex].\r\n

    Solution<\/h4>\r\nSubstitute the slope (m) into\u00a0[latex]y=mx+b[\/latex]:\r\n

    [latex]y=3x+b[\/latex]<\/p>\r\nSubstitute the point [latex](1,4)[\/latex] for [latex]x[\/latex] and [latex]y[\/latex]:\r\n

    [latex]4=3\\left(1\\right)+b[\/latex]<\/p>\r\nSolve for b:\r\n

    [latex]\\begin{array}{l}4=3+b\\\\1=b\\end{array}[\/latex]<\/p>\r\nWrite the equation of the line [latex]y=mx+b[\/latex]\u00a0with [latex]m=3[\/latex]\u00a0and [latex]b=1[\/latex].\r\n

    Answer<\/h4>\r\n[latex]y=3x+1[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nTo confirm our algebra, we can check by graphing the equation [latex]y=3x+1[\/latex]. The equation checks because when graphed it passes through the point [latex](1,4)[\/latex].\r\n
    \"An<\/div>\r\n \r\n
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    Example<\/h3>\r\nDetermine the equation of the line that has a slope of [latex]\\frac{7}{8}[\/latex]\u00a0and contains the point [latex]\\left(4,\\frac{5}{4}\\right)[\/latex].\r\n

    Solution<\/h4>\r\nSubstitute the slope (m) into [latex]y=mx+b[\/latex]:\r\n

    [latex]\\begin{array}{l}y=mx+b\\\\\\\\y=\\frac{7}{8}x+b\\end{array}[\/latex]<\/p>\r\nSubstitute the point [latex]\\left(4,\\frac{5}{4}\\right)[\/latex]\u00a0for x and y:\r\n

    [latex]\\frac{5}{4}=\\frac{7}{8}\\left(4\\right)+b[\/latex]<\/p>\r\nSolve for b:\r\n

    [latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\frac{5}{4}=\\frac{28}{8}+b\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\frac{5}{4}=\\frac{14}{4}+b\\\\\\\\\\frac{5}{4}-\\frac{14}{4}=\\frac{14}{4}-\\frac{14}{4}+b\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-\\frac{9}{4}=b\\end{array}[\/latex]<\/p>\r\nWrite the equation of the line [latex]y=mx+b[\/latex] with [latex] \\displaystyle m=\\frac{7}{8}[\/latex] and [latex] \\displaystyle b=-\\frac{9}{4}[\/latex].\r\n

    Answer<\/h4>\r\n[latex]y=\\frac{7}{8}x-\\frac{9}{4}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
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    Try It<\/h3>\r\nWrite the equation of the line that has a slope of [latex]\\frac{3}{2}[\/latex]\u00a0and contains the point [latex]\\left(2,5\\right)[\/latex].\r\n\r\n[reveal-answer q=\"hjm698\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm698\"]\r\n\r\n[latex]y=\\frac{3}{2}x+2[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n \r\n

    Watch the video below for another example of how to find the equation given the slope and a point on the line.<\/p>\r\n