Working together - math question

Let's say Jack's work rate is (1 job)/(k hours).

Let's say Jill's work rate is (1 job)/(j hours).

Time in which Jill would do all the work alone = j hours.

Jack worked 2/3 of that time, or (2/3)j.

Amount Jack completed = rate * time = (1/k) * (2/3)j = 2j/3k.

Amount Jack left uncompleted, which Jill had to do: 1 – (2j/3k).

Time it took Jill to do that portion: time = amount/rate = (1 – (2j/3k))/(1/j).

Time combined was 2 hours more than if they had worked together:

(2/3)j + (1 – (2j/3k))/(1/j) = 2 + time if working together

Time if working together would be: combined rate * time = 1 job

(1/k + 1/j) * time = 1

time if working together = 1/(1/k + 1/j) = 1/(j + k)/jk = jk/(j + k)

(2/3)j + (1 – (2j/3k))/(1/j) = 2 + jk/(j + k)

Amount of work Jack would have done if working together:

amount = rate * time = (1/k) * jk/(j + k) = j/(j + k)

That would be half as much as what he actually left for Jill:

j/(j + k) = 1/2 * (1 – (2j/3k))

I suspect the two bold equations are what's needed to find j and k. It's two equations in two variables.