Slope is defined as:

\(\displaystyle m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}\)

If we let the second point be the arbitrary point \((x,y)\), we then have:

\(\displaystyle m=\frac{y-y_1}{x-x_1}\)

This gives us the locus of all points that have a slope of \(m\) with respect to \(\left(x_1,y_1\right)\), and it can be arranged as:

\(\displaystyle y-y_1=m\left(x-x_1\right)\)