Dr. Lum teaches part-time at two different community colleges, Hilltop College and Serra College. Dr. Lum can teach up to 5 classes per semester. For every class taught by him at Hilltop College, he needs to spend 3 hours per week preparing lessons and grading papers, and for each class at Serra College, he must do 4 hours of work per week. He has determined that he cannot spend more than 18 hours per week preparing lessons and grading papers. If he earns $4,000 per class at Hilltop College and $4,200 per class at Serra College, how many classes should he teach at each college to maximize his income, and what will be 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.