Lines can be written in the form , where is the gradient and is the intercept.
So we could write the equation of two lines as
You should know that parallel lines never cross. These two lines will cross when their equations are equal. So
Obviously there is no solution when the denominator is , in other words, where .
So the only time when there will not be a point of intersection is when the two gradients are equal.
In other words, the lines are parallel when their gradients are equal.