Dr. Lum teaches part-time at two community colleges, Hilltop College and Serra College. Dr. Lum can teach up to 2 classes per semester. For every class he teaches at Hilltop College, he must do 8 hours of work per week. For each class at Serra College, he must do 13 hours of work per week. He has determined that he cannot spend more than 21 hours per week working. If he earns $6000 per class at Hilltop College and $7000 per class at Serra College, how many classes should he teach at each college to maximize his income?

Equation 1: Dr. Lum can teach up to 5 classes per semester at both colleges. If 'h' represents the number of classes taken at Hilltop College, and 's', the number of classes taken at Serra College,

Equation 1 becomes: h + s = 5

Equation 2: Dr. Lum spends 3 hours per week preparing for Hilltop classes, and 4 hours for Serra classes, subject to a maximum of 18 hours per week.

Equation 2 becomes: 3h + 4s = 18

Now solving using substitution method,

We can derive an equation 3 from equation 1, as follows.

Equation 3: h = 5 - s.

Substituting equation 3 into equation 2, equation 2 becomes

3(5 - s) + 4s = 18

= 15 - 3s + 4s = 18

= 15 + s = 18

= s = 3.

With s = 3, using equation 3, h = 5 - 3 = 2.

Therefore, Dr. Lum should teach 2 classes at Hilltop, and 3 classes at Serra.

His income will be

$4,000 (2) + $4,200 (3)

= $20,600.