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

and .

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.