https://file.aiquickdraw.com/mathbot/file/17286670616367tzg2dzk.jpg

The problem shown in the image is in Gujarati and translates to:

**"Find the common divisor whose sum is \(7 + \sqrt{5}\) and difference is \(7 - \sqrt{5}\)."**

This is asking for the common values or expressions that have both a sum and a difference related to the given numbers.

The approach here is to apply the system of equations. Let's denote the two unknown numbers as \(a\) and \(b\).

From the question, we are given two equations:
1. \( a + b = 7 + \sqrt{5} \)
2. \( a - b = 7 - \sqrt{5} \)

### Solving the system of equations:
To find \(a\) and \(b\), we can add and subtract these equations.

**Step 1:** Add the two equations:
\[
(a + b) + (a - b) = (7 + \sqrt{5}) + (7 - \sqrt{5})
\]
\[
2a = 14 \quad \Rightarrow \quad a = \frac{14}{2} = 7
\]

**Step 2:** Subtract the second equation from the first:
\[
(a + b) - (a - b) = (7 + \sqrt{5}) - (7 - \sqrt{5})
\]
\[
2b = 2\sqrt{5} \quad \Rightarrow \quad b = \frac{2\sqrt{5}}{2} = \sqrt{5}
\]

### Final Answer:
The two numbers are \( a = 7 \) and \( b = \sqrt{5} \).

Would you like more details on this solution, or have any specific questions? Here are some related questions that may help you:

1. Can you solve similar equations for different values of sums and differences?
2. How can you apply the same method to more complex equations?
3. What happens if the square roots involve other numbers, not just \(\sqrt{5}\)?
4. How would you solve this if both numbers involved square roots?
5. Can this method be extended to three or more variables?

**Tip:** Always double-check the operations when adding and subtracting square roots to avoid errors in simplifying expressions.

Find the common divisor whose sum is 7 + √5 and difference is 7 - √5.

Learn how to solve a system of linear equations where the sum of two numbers is 7 + √5 and their difference is 7 - √5. This problem involves key algebraic concepts, including solving simultaneous equations and simplifying square roots. Suitable for students in Grades 9-10.

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