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Solution:

Given:
- Total number of students = 35
- Number of students who have taken mathematics but not economics = 17

Let's denote:
- Number of students who have taken both mathematics and economics as "x"
- Number of students who have taken economics but not mathematics as "y"

Step 1:
We can represent the given information in a Venn diagram as shown below.



Step 2:
From the Venn diagram, we can write the following equations:

- Number of students who have taken mathematics = 17 + x
- Number of students who have taken economics = x + y
- Total number of students = 17 + x + y + (students who have taken neither)

Since each student has taken mathematics or economics or both, we know that the total number of students who have taken either mathematics or economics is equal to the total number of students.

Therefore, we can write:

- Number of students who have taken either mathematics or economics = 17 + x + y

Step 3:
Substituting the equations from Step 2 into the equation above, we get:

- 17 + x + y = 35

Simplifying, we get:

- x + y = 18

Step 4:
Using the Venn diagram and the equations from Step 2, we can solve for "x" and "y" as follows:

- Number of students who have taken both mathematics and economics (x) = (Number of students who have taken mathematics) - (Number of students who have taken mathematics but not economics)
= (17 + x) - 17
= x

Therefore, x = Number of students who have taken both mathematics and economics = 9

- Number of students who have taken economics but not mathematics (y) = (Number of students who have taken economics) - (Number of students who have taken both mathematics and economics)
= (x + y) - x
= y

Therefore, y = Number of students who have taken economics but not mathematics = 9

Step 5:
The final answer is:

- Number of students who have taken both mathematics and economics = 9
- Number of students who have taken economics but not mathematics = 9

Therefore, there are 9 students who have taken both mathematics and economics, and 9 students who have taken economics but not mathematics.