Question

what is the equation of the line which includes points (1,1) and (1,5)

Answer 1

\(\bf \begin{array}{lllll}
&x_1&y_1&x_2&y_2\\
&({\color{red}{ 1}}\quad ,&{\color{blue}{ 1}})\quad &({\color{red}{ 1}}\quad ,&{\color{blue}{ 5}})
\end{array}
\\\quad \\

slope = {\color{green}{ m}}= \cfrac{rise}{run} \implies \cfrac{{\color{blue}{ y_2}}-{\color{blue}{ y_1}}}{{\color{red}{ x_2}}-{\color{red}{ x_1}}}
\\ \quad \\

y-{\color{blue}{ y_1}}={\color{green}{ m}}(x-{\color{red}{ x_1}})\qquad \textit{plug in the values and solve for "y"}\\
\qquad \uparrow\\
\textit{point-slope form}\)

Answer 2

This is a special case where the slope is infinite. This is really the same thing as just a line were all the x-values are exactly the same, so x=1 is the equation of your line.

Similarly, the line y=1 has a slope of zero. So you can sort of see how this is also a line but perpendicular.