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The system of equations presented is:

\[
\begin{cases}
y = x + 1 \\
y^2 + x^2 = 25
\end{cases}
\]

### Step-by-Step Solution Using Substitution:
1. **Substitute \( y = x + 1 \) into the second equation**:

   The second equation is:
   \[
   y^2 + x^2 = 25
   \]

   Substitute \( y = x + 1 \):
   \[
   (x + 1)^2 + x^2 = 25
   \]

2. **Simplify the equation**:

   Expand \( (x + 1)^2 \):
   \[
   x^2 + 2x + 1 + x^2 = 25
   \]

   Combine like terms:
   \[
   2x^2 + 2x + 1 = 25
   \]

   Subtract 25 from both sides:
   \[
   2x^2 + 2x - 24 = 0
   \]

   Simplify by dividing by 2:
   \[
   x^2 + x - 12 = 0
   \]

3. **Solve the quadratic equation** \( x^2 + x - 12 = 0 \):

   Factor the quadratic:
   \[
   (x + 4)(x - 3) = 0
   \]

   Thus, the solutions for \( x \) are:
   \[
   x = -4 \quad \text{and} \quad x = 3
   \]

4. **Find the corresponding \( y \)-values**:

   Using \( y = x + 1 \):
   - For \( x = -4 \): \( y = -4 + 1 = -3 \)
   - For \( x = 3 \): \( y = 3 + 1 = 4 \)

5. **Solutions**:
   
   The system has two solutions:
   \[
   (-4, -3) \quad \text{and} \quad (3, 4)
   \]

### Method Choice Explanation
- **Substitution** was chosen because the first equation is already solved for \( y \), making it easy to substitute into the second equation.
- A **graphical method** would also be useful to visually confirm the intersection points of the circle \( y^2 + x^2 = 25 \) and the line \( y = x + 1 \).

Would you like more details or have any other questions? 

Here are 5 related questions for further practice:

1. What is the solution of the system if the line equation was \( y = 2x + 1 \) instead?
2. How do you graphically find the intersection points of a circle and a line?
3. Can you describe the nature of the solutions when a line intersects a circle at exactly one point?
4. What happens if the system involves an ellipse instead of a circle?
5. How does changing the radius of the circle affect the solutions?

**Tip**: Always check your solutions by substituting back into the original equations to verify their correctness!

You are given the following system of equations. Solve the system by substitution or by a graphical method, and explain how to choose a method:

y = x + 1

y^2 + x^2 = 25

Learn how to solve the system of equations y = x + 1 and y^2 + x^2 = 25 using the substitution method. This problem involves key algebraic and geometric concepts, including quadratic equations and the properties of circles, suitable for Grades 9-11.

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