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The image contains a math problem where the objective is to find one of the values of \( x \) by solving the given system of equations. The system is expressed as follows:

\[
\frac{\sqrt{7}}{x} + \frac{\sqrt{5}}{x} = \sqrt{2}
\]

Let's solve this step by step.

### Step 1: Combine the fractions on the left side.
The equation can be rewritten as:

\[
\frac{\sqrt{7} + \sqrt{5}}{x} = \sqrt{2}
\]

### Step 2: Isolate \( x \).
To isolate \( x \), multiply both sides of the equation by \( x \) and then divide by \( \sqrt{2} \):

\[
x = \frac{\sqrt{7} + \sqrt{5}}{\sqrt{2}}
\]

### Step 3: Rationalize the denominator.
To simplify the expression, multiply the numerator and the denominator by \( \sqrt{2} \) to rationalize the denominator:

\[
x = \frac{(\sqrt{7} + \sqrt{5}) \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{\sqrt{14} + \sqrt{10}}{2}
\]

### Step 4: Check the options.
The options given in the problem are:

A) 16  
B) 7  
C) 8  
D) 12  
E) 15

We need to check which one of these is closest to \( \frac{\sqrt{14} + \sqrt{10}}{2} \).

### Step 5: Approximate the square roots.
Let's approximate the values:

\[
\sqrt{14} \approx 3.74 \quad \text{and} \quad \sqrt{10} \approx 3.16
\]

\[
\text{So, } \frac{\sqrt{14} + \sqrt{10}}{2} \approx \frac{3.74 + 3.16}{2} = \frac{6.9}{2} \approx 3.45
\]

Since none of the given options are close to 3.45, it seems there might be an issue. After reconsideration, it appears there could have been a mistake in the approximation or interpretation of the problem. 

Given the options, it's likely that the square roots involve more precise values, or the problem might have an implied assumption we're missing (e.g., the value might need to be squared).

Without additional context or assumptions, the closest choice would not fit well, which suggests the problem might require specific educational techniques.

### Final Answer:
Considering the problem as stated and a more accurate understanding or re-evaluation might be necessary for an exact match to the options provided.

---

Would you like further clarification or additional details on any step? Here are some related questions to expand your understanding:

1. How do you properly rationalize denominators in fractions involving square roots?
2. What are some common approximations of square roots used in competitive exams?
3. How do you simplify expressions involving multiple square roots?
4. Can you find alternative methods to solve equations involving square roots?
5. What strategies are best when the solution isn't apparent from the given options?

**Tip:** Always recheck calculations, especially when approximate values are used, to ensure they match the context of the problem.

Learn how to solve equations involving square roots with a detailed step-by-step guide. This problem focuses on rationalizing denominators and approximating square roots. Suitable for advanced high school students.

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