The 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 4,700 students who would like to take a course, and there are 58 professors to teach them. Sections of Ancient History have 100 students each, sections of Medieval History hold 50 students each, and sections of Modern History have 200 students each. Modern History sections are taught by a team of two professors, while Ancient and Medieval History need only one professor per section. How many sections of each course should the chair schedule in order to offer all the sections that they are allowed, accommodate all of the students, and give one teaching assignment to each professor?

The number of professors is 58 where Ancient and Medieval History need only 1 professor per section and Modern History sections are taught by a team of 2 professors, so

[tex]x+y+2z=58[/tex]

Solving the first and last equations by subtracting them we get

[tex]-z=-13\\\Rightarrow z=13[/tex]

Substituting this value in the second equation and first equation we get

[tex]100x+50y+2600=4700\\\Rightarrow 100x+50y=2100\\\Rightarrow 2x+y=42[/tex]

[tex]x+y+13=45\\\Rightarrow x+y=32[/tex]

Solving these equations by subtracting them we get

[tex]x=10[/tex]

[tex]x+y=32\\\Rightarrow y=32-10=22[/tex]

So, [tex]x=10, y=22, z=13[/tex]

Hence, the number of sections in Ancient history is 10, number of students in Medieval history is 22 and the number of students in Modern history is 13.