Learning Outcomes
Slope-Intercept FormPerhaps the most familiar form of a linear equation is slope-intercept form written as [latex]y=mx+b[/latex], where [latex]m=\text{slope}[/latex] and [latex]b=y\text{-intercept}[/latex]. Let us begin with the slope. The Slope of a LineThe slope of a line refers to the ratio of the vertical change in y over the horizontal change in x 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. [latex]m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[/latex] 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]. A General Note: The Slope of a LineThe slope of a line, m, represents the change in y over the change in x. 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: [latex]m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[/latex] Example: Finding the Slope of a Line Given Two PointsFind the slope of a line that passes through the points [latex]\left(2,-1\right)[/latex] and [latex]\left(-5,3\right)[/latex]. Try ItFind the slope of the line that passes through the points [latex]\left(-2,6\right)[/latex] and [latex]\left(1,4\right)[/latex]. Example: Identifying the Slope and y-intercept of a Line Given an EquationIdentify the slope and y-intercept given the equation [latex]y=-\frac{3}{4}x - 4[/latex]. The Point-Slope FormulaGiven the slope and one point on a line, we can find the equation of the line using point-slope form. [latex]y-{y}_{1}=m\left(x-{x}_{1}\right)[/latex] This is an important formula, as it will be used in other areas of College Algebra and often in Calculus to find the equation of a tangent line. We need only one point and the slope of the line to use the formula. After substituting the slope and the coordinates of one point into the formula, we simplify it and write it in slope-intercept form. A General Note: The Point-Slope FormulaGiven one point and the slope, using point-slope form will lead to the equation of a line: [latex]y-{y}_{1}=m\left(x-{x}_{1}\right)[/latex] Example: Finding the Equation of a Line Given the Slope and One PointWrite the equation of the line with slope [latex]m=-3[/latex] and passing through the point [latex]\left(4,8\right)[/latex]. Write the final equation in slope-intercept form. Try ItGiven [latex]m=4[/latex], find the equation of the line in slope-intercept form passing through the point [latex]\left(2,5\right)[/latex]. Example: Finding the Equation of a Line Passing Through Two Given PointsFind the equation of the line passing through the points [latex]\left(3,4\right)[/latex] and [latex]\left(0,-3\right)[/latex]. Write the final equation in slope-intercept form. Standard Form of a LineAnother way that we can represent the equation of a line is in standard form. Standard form is given as [latex]Ax+By=C[/latex] where [latex]A[/latex], [latex]B[/latex], and [latex]C[/latex] are integers. The x and y-terms are on one side of the equal sign and the constant term is on the other side. Example: Finding the Equation of a Line and Writing It in Standard FormFind the equation of the line with [latex]m=-6[/latex] and passing through the point [latex]\left(\frac{1}{4},-2\right)[/latex]. Write the equation in standard form. Try ItFind the equation of the line in standard form with slope [latex]m=-\frac{1}{3}[/latex] which passes through the point [latex]\left(1,\frac{1}{3}\right)[/latex]. Vertical and Horizontal LinesThe equations of vertical and horizontal lines do not require any of the preceding formulas, although we can use the formulas to prove that the equations are correct. The equation of a vertical line is given as [latex]x=c[/latex] where c is a constant. The slope of a vertical line is undefined, and regardless of the y-value of any point on the line, the x-coordinate of the point will be c. Suppose that we want to find the equation of a line containing the following points: [latex]\left(-3,-5\right),\left(-3,1\right),\left(-3,3\right)[/latex], and [latex]\left(-3,5\right)[/latex]. First, we will find the slope. [latex]m=\frac{5 - 3}{-3-\left(-3\right)}=\frac{2}{0}[/latex] Zero in the denominator means that the slope is undefined and, therefore, we cannot use point-slope form. However, we can plot the points. Notice that all of the x-coordinates are the same and we find a vertical line through [latex]x=-3[/latex]. The equation of a horizontal line is given as [latex]y=c[/latex] where c is a constant. The slope of a horizontal line is zero, and for any x-value of a point on the line, the y-coordinate will be c. Suppose we want to find the equation of a line that contains the following set of points: [latex]\left(-2,-2\right),\left(0,-2\right),\left(3,-2\right)[/latex], and [latex]\left(5,-2\right)[/latex]. We can use point-slope form. First, we find the slope using any two points on the line. [latex]\begin{array}{l}m=\frac{-2-\left(-2\right)}{0-\left(-2\right)}\hfill \\ =\frac{0}{2}\hfill \\ =0\hfill \end{array}[/latex] Use any point for [latex]\left({x}_{1},{y}_{1}\right)[/latex] in the formula, or use the y-intercept. [latex]\begin{array}{l}y-\left(-2\right)=0\left(x - 3\right)\hfill \\ y+2=0\hfill \\ y=-2\hfill \end{array}[/latex] The graph is a horizontal line through [latex]y=-2[/latex]. Notice that all of the y-coordinates are the same. The line x = −3 is a vertical line. The line y = −2 is a horizontal line. try itUse an online graphing tool to graph the following:
Example: Finding the Equation of a Line Passing Through the Given PointsFind the equation of the line passing through the given points: [latex]\left(1,-3\right)[/latex] and [latex]\left(1,4\right)[/latex]. Try ItFind the equation of the line passing through [latex]\left(-5,2\right)[/latex] and [latex]\left(2,2\right)[/latex]. Contribute!Did you have an idea for improving this content? We’d love your input. Improve this pageLearn More How do you find the equation of a vertical and horizontal line?Horizontal lines go left and right and are in the form of y = b, where b represents the y-intercept, while vertical lines go up and down and are in the form of x = a where a represents the shared x-coordinate of all points.
How do you find the vertical line that passes through a point?Vertical lines
If two points have the same x -coordinates, only a vertical line can pass through both points. Each point on a vertical line has the same x -coordinate. If two points have the same x -coordinate, c , the equation of the line is x=c . The x -intercept of a vertical line x=c is the point (c,0) .
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