What exactly is the slope of a line?
The slope of a line is the rate at which the value of the y-coordinate increases with respect to an increase in the x-coordinate. With respect to the angle made by the line with the positive x-axis, the slope is the tangent of the angle made.
As the slope of a line increases, the difference between the values of the y-coordinate of two points increases when compared to the change in the value of the x-coordinate. For example the line y = 4 has a slope of 0 as there is no change in the y-coordinate for a change in the x-coordinate. x - y = 0 has a slope of 1 as the y-coordinate changes at the same rate as the x-coordinate. On the other extreme a vertical line like x = 8 has a slope equal to infinity.