Slope (or gradient) is used to determine if lines are parallel, as parallel lines share the same gradient.
The gradient is used to determine whether they intersect, as you'll have to plug x and y values in to both of the line's equations, which involves having the line's gradients.
The gradient is used to see if points are collinear, as you need to determine whether the gradient between two points is equal to the gradient of another 2 points, and if so, the points are collinear.
Finally, the gradient is used to find the equation of the line, as we require to plug in the gradient in to equations such as (y - y1 = m(x - x1)) in order to find the equation of the line.