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.