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You have provided two equations:

1. \( x + y = 7 \)
2. \( x^2 + y^2 = 25 \)

We need to find the coordinates \((x, y)\) that satisfy both equations. Let's solve this system of equations.

### Steps:

1. From the first equation, solve for \( y \):
   \[
   y = 7 - x
   \]
   
2. Substitute \( y = 7 - x \) into the second equation:
   \[
   x^2 + (7 - x)^2 = 25
   \]

3. Expand the equation:
   \[
   x^2 + (49 - 14x + x^2) = 25
   \]
   Simplify:
   \[
   2x^2 - 14x + 49 = 25
   \]

4. Subtract 25 from both sides:
   \[
   2x^2 - 14x + 24 = 0
   \]
   Simplify:
   \[
   x^2 - 7x + 12 = 0
   \]

5. Solve this quadratic equation using the quadratic formula:
   \[
   x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(12)}}{2(1)}
   \]
   Simplify:
   \[
   x = \frac{7 \pm \sqrt{49 - 48}}{2} = \frac{7 \pm 1}{2}
   \]
   Thus, the solutions for \( x \) are:
   \[
   x = \frac{7 + 1}{2} = 4 \quad \text{or} \quad x = \frac{7 - 1}{2} = 3
   \]

6. Substitute these values of \( x \) back into the equation \( y = 7 - x \) to find the corresponding \( y \) values:
   - If \( x = 4 \), then \( y = 7 - 4 = 3 \)
   - If \( x = 3 \), then \( y = 7 - 3 = 4 \)

### Final Answer:
The two sets of coordinates that satisfy both equations are:
\[
(4, 3) \quad \text{and} \quad (3, 4)
\]

Would you like further details or have any questions?

Here are 5 related questions to expand on this topic:
1. How can this system of equations be solved graphically?
2. Can we generalize this method for any similar pair of equations?
3. What is the geometric interpretation of the equations \(x + y = 7\) and \(x^2 + y^2 = 25\)?
4. How does the quadratic formula work, and what is its derivation?
5. What are other methods for solving systems of non-linear equations?

**Tip**: Always check both possible solutions by substituting them back into the original equations to verify their correctness.

Find the coordinates of the general solution for the system of equations: x + y = 7 and x^2 + y^2 = 25.

Learn how to find the coordinates of the general solution for the system of equations x + y = 7 and x^2 + y^2 = 25. This solution involves algebraic methods like substitution and solving quadratic equations. Suitable for students in Grades 8-10.

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