Notes on Writing Equations of a Line in Slope-Intercept Form

4.1 Writing Equations of a Line in Slope-Intercept Form (y=mx+b)

Objective:

  • Write an equation of a line given:-

    • Slope & y-intercept

    • Two points

Background Info:

  • To write an equation in slope-intercept form, you need the slope (mm) and y-intercept (bb).

  • The slope-intercept form is given by: y=mx+by = mx + b - Where: - yy is the dependent variable
    - mm is the slope
    - xx is the independent variable
    - bb is the y-intercept

Writing an Equation Given Slope & Y-Intercept
Sample:

  • Write the equation of a line when the slope is -1 and the y-intercept is 3.

    • Given: - Slope (mm) = -1

      • Y-intercept (bb) = 3

    • Using the slope-intercept form: y=mx+by = mx + b

    • Substitute the given values: y=−1x+3y = -1x + 3

    • Therefore, the equation of the line is: y=−x+3y = -x + 3

Practice

  1. - Slope = 2

    • Y-int = -4

    • __

  2. - Slope = 0

    • Y-int = 7

    • __

  3. - Slope = 3

    • Y-int = 0

    • __

  4. - Slope = −13\frac{-1}{3}

    • Y-int = -1

    • __

Writing an Equation When Given Two Points (One Will Be the Y-Intercept)
Sample:

  • Given two points: (4,1) and (0,3)

  • Note: (0,3) is the y-intercept because x=0

  • The y-intercept (bb) = 3

  • Calculate the slope (mm) using the formula: m=y2−y1x2−x1m = \frac{y2 - y1}{x2 - x1}

  • m=3−10−4=2−4=−12m = \frac{3 - 1}{0 - 4} = \frac{2}{-4} = \frac{-1}{2}

  • Using the slope-intercept form: y=mx+by = mx + b

  • Substitute the values: y=−12x+3y = \frac{-1}{2}x + 3

Sample

  • (0, -2) and (3, 2)

  • Since the first point is (0, -2), the y-intercept (bb) = -2

  • Calculate the slope (mm):- m=y2−y1x2−x1m = \frac{y2 - y1}{x2 - x1}

    • m=2−(−2)3−0=43m = \frac{2 - (-2)}{3 - 0} = \frac{4}{3}

  • So the equation is:- y=43x−2y = \frac{4}{3}x - 2

Sample

  • (4, 1) and (0, 3)

  • Since the second point is (0, 3), it is a y-intercept

    • Y-int b=3b = 3

  • Calculate the slope from the graph

    • m=riserun=−24=−12m = \frac{rise}{run} = \frac{-2}{4} = \frac{-1}{2}

  • So the equation is:- y=−12x+3y = \frac{-1}{2}x + 3

Sample

  • (-3, -2) and (0, 1)

  • Since the second point is (0, 1), it is a y-intercept

    • Y-int b=1b = 1

- m=riserunm = \frac{rise}{run}, from (-3, -2) to (0, 1), the rise is 3 and the run is 3, so m=33=1m = \frac{3}{3} = 1