Systems of Equation application problem, regarding student/professor/sections per cla

Question reads

Enormous state university history department offers three courses, Ancient , medieval, and modern history, and the chairperson is trying to decide how many sections of each to offer this semester. the department is allowed to offer 45 sections total, there are 5000 students who would like to take a course, and there are 60 professor to teach them. Sections of ancient history have 100 students each, medieval hold 50 students, and modern history has 200 students each. Modern history is thought by 2 professor, while ancient and medieval history need only 1 professor per section.

How many sections of each course should the chair schedule in order to offer all the sections that they are allowed to, accommodate all students, and give one teaching assignment to each professor?

i set up my table as

_AH_.MeH_MoH__Total

Student per section| 100 | 50 | 200 | 5000

Professor..............| 1 | 1 | 2 | 60

Sections...............|0.....|0 | 0 | 45

my matrices looked like

as you can see, i likely messed up at the "sections" part.

I ended up with

alternatively:

x = (400-300z)/100

y= (1000 + 100z)/50

z=z

but anyways my point is nether of the previous 2 gets to the proper answer, most likely because of "section" consisting of 3 zeros.