Question

In a class of 38 students each student study at least once of the following subjects French mathematics and history the following stable give further details of the subject studies French only 9 mathematics 8 history honey one French and mathematics 7 French and history 9 history and mathematics 8
A if x is the number of students who state all the three subjects illustrate the above information in a venn diagram
ii. write an equation in x involving the number of students in the region of the diagram

Answer 1

To analyze the problem, we can define the sets based on the subjects studied by the students in the class. Let's denote:

- \( F \) for the set of students studying French
- \( M \) for the set of students studying Mathematics
- \( H \) for the set of students studying History

We can denote:

- \( |F| \) = number of students studying French
- \( |M| \) = number of students studying Mathematics
- \( |H| \) = number of students studying History

From the information provided, we can summarize the students studying only one subject and those studying combinations of subjects as follows:

1. Students studying only French: \( |F| - (FM + FH + FMH) = 9 \)
2. Students studying only Mathematics: \( |M| - (FM + HM + FMH) = 8 \)
3. Students studying only History: \( |H| - (FH + HM + FMH) = 1 \)
4. Students studying both French and Mathematics but not History: \( |FM| = 7 \)
5. Students studying both French and History but not Mathematics: \( |FH| = 9 \)
6. Students studying both History and Mathematics but not French: \( |HM| = 8 \)
7. Students studying all three subjects: \( |FMH| = x \)

Now, we also know that the total number of students studying at least one subject is:
\[
|F \cup M \cup H| = 38
\]

Let's put this information into a Venn diagram representation and create an equation.

### Venn Diagram Set Notation

We can denote:
- \( |F \cap M| = 7 + x \) (French and Mathematics, including those studying all three)
- \( |F \cap H| = 9 + x \) (French and History, including those studying all three)
- \( |M \cap H| = 8 + x \) (Mathematics and History, including those studying all three)

Using the principle of inclusion-exclusion for three sets:
\[
|F \cup M \cup H| = |F| + |M| + |H| - |F \cap M| - |F \cap H| - |M \cap H| + |F \cap M \cap H|
\]

Substituting what we have, we can express the total number of students as:
\[
38 = (9 + (7 + x) + (9 + x) + (8 + x)) - (7 + x) - (9 + x) - (8 + x) + x
\]

### Setting up the equation

Simplifying the equation:
\[
38 = (9 + 7 + 9 + 8 + 2x) - (7 + 9 + 8 + 2x) + x
\]
\[
= 33 + 2x - 24 - 2x + x
\]
\[
= 33 - 24 + x
\]
\[
= 9 + x
\]

Thus, our equation would be:
\[
9 + x = 38
\]

Taking \( x \) to one side gives:
\[
x = 38 - 9
\]
\[
x = 29
\]

### Final Answer
1. The value of \( x \) is 29, meaning 29 students are studying all three subjects.
2. The equation that illustrates the relationship in terms of \( x \) is:
\[
38 = 9 + 7 + 8 + 1 + 9 + 8 + x
\]

### Note
Please double-check calculations if you re-evaluate combinations to ensure we classify students in various subject categories correctly. It’s also important to visualize through a Venn diagram for clarity on student representation in each intersection.