Question 1202198
you can use the gauss-jordan matrix calculator to solve this.
<a href = "https://matrix.reshish.com/gauss-jordanElimination.php" target = "_blank">https://matrix.reshish.com/gauss-jordanElimination.php</a>
you set up a 3 row by 4 column matrix.
matric entries are:
<pre>
                   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
</pre>
run the program and here are the results.


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


<img src = "http://theo.x10hosting.com/2023/050903.jpg">


<img src = "http://theo.x10hosting.com/2023/050904.jpg">


<img src = "http://theo.x10hosting.com/2023/050905.jpg">


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:
<pre>
                      ancient    medieval    modern     total
sections                 8          22         15         45
teachers                 8          22         30         60
students               800        1100       3000       4900   
</pre>
if you need to know how to do it manually, let me know.
theo