SOLUTION: 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 t

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Question 1202198: 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 4,900 students who would like to take a course, and there are 60 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? HINT [See Example 2.] (Assume each section has the maximum number of students allowed.)
Ancient History sections Medieval History sections Modern History sections
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
you can use the gauss-jordan matrix calculator to solve this.
https://matrix.reshish.com/gauss-jordanElimination.php
you set up a 3 row by 4 column matrix.
matric entries are:
                   column 1      column 2     column 3     column 4
variable              x1           x2           x3            
row 1                  1            1            1            45
row 2                  1            1            2            60
row 3                100           50          200          4900

run the program and here are the results.

first you set the matrix
next you input the numbers.
next it calculates the result.







x1 is the number of ancient history sections.
x2 is the number of medieval history sections.
x3 is the number of modern history sections.

your equations are:

number of sections:
1*x1 + 1*x2 + 1*x3 = 45

number of professors per section:
1*x1 + 1*x2 + 2*x3 = 60

number of students per section:
100*x1 + 50*x2 + 200*x3 = 4900

your solution is:
8 sections of ancient history.
22 sections of medieval history
15 sections of modern history.

your answer matrix is:
                      ancient    medieval    modern     total
sections                 8          22         15         45
teachers                 8          22         30         60
students               800        1100       3000       4900

if you need to know how to do it manually, let me know.
theo