1.14.2 Writing Equations for Point-Slope and Slope-Intercept

Method 1: Using Point-Slope Form

Find an equation of a line that contains the points $(5,4)$ and $(3,6)$. Write the equation in slope-intercept form.

Step 1 - Find the slope between the two points.

$m=\frac{6-4}{3-5}=\frac{2}{-2}=-1$

Step 2 - Choose a point to substitute into point-slope form.

$y-y_1=m(x-x_1)$

$y-4=-1(x-5)$ OR $y-6=-1(x-3)$

Step 3 - Write the equation in slope-intercept form.

$\begin{array}{ccc}\hfill y-4& \hfill =\hfill & -1(x-5)\hfill \\ \hfill y-4& \hfill =\hfill & -1x+5\hfill \\ \hfill y& \hfill =\hfill & -x+9\hfill \end{array}$

OR

$\begin{array}{ccc}\hfill y-6& \hfill =\hfill & -1(x-3)\hfill \\ \hfill y-6& \hfill =\hfill & -1x+3\hfill \\ \hfill y& \hfill =\hfill & -x+9\hfill \end{array}$

Method 2: Writing an Equation in Slope-Intercept Form by finding $b$.

Find the equation of the line containing the points $(3,1)$ and $(5,6)$ in slope-intercept form.

Step 1 - Find the slope between the two points.

$m=\frac{6-1}{5-3}=\frac{5}{2}$

Step 2 - Choose a point to substitute into slope-intercept form then solve for $b$.

$y=mx+b$

$\begin{array}{ccc}\hfill y& \hfill =\hfill & \frac{5}{2}x+b\hfill \\ \hfill 1& \hfill =\hfill & \frac{5}{2}(3)+b\hfill \\ \hfill \frac{2}{2}& \hfill =\hfill & \frac{15}{2}+b\hfill \\ \hfill b& \hfill =\hfill & \frac{13}{2}\hfill \end{array}$

OR

$\begin{array}{ccc}\hfill y& \hfill =\hfill & \frac{5}{2}x+b\hfill \\ \hfill 6& \hfill =\hfill & \frac{5}{2}(5)+b\hfill \\ \hfill \frac{12}{2}& \hfill =\hfill & \frac{25}{2}+b\hfill \\ \hfill b& \hfill =\hfill & \frac{13}{2}\hfill \end{array}$

Step 3 - Write the equation in slope-intercept form.

$y=\frac{5}{2}x+\frac{13}{2}$