determine the number of solutions for x and y for each of the following scenarios. justify your answer for each case.

case 1: c<0
case 2: c=0
case 3: c>0

The problem you presented seems to be about determining the number of solutions for \(x\) and \(y\) in a given system of equations. However, you didn't specify the system or the equation involving \(c\). Given this, I'll assume you are referring to a common scenario, such as finding the number of solutions to a quadratic equation in two variables, or perhaps a linear system. 

Let's proceed with the assumption that we are dealing with a quadratic equation of the form:

\[
ax^2 + by^2 = c
\]

### Case 1: \( c < 0 \)

For the equation \( ax^2 + by^2 = c \), when \( c < 0 \):

- **Justification**: \( ax^2 \) and \( by^2 \) are both non-negative because squares of real numbers are non-negative (\( ax^2 \geq 0 \) and \( by^2 \geq 0 \)). Therefore, the sum \( ax^2 + by^2 \) is always non-negative. If \( c < 0 \), there's no way to satisfy the equation because the left-hand side will always be non-negative, while the right-hand side is negative.
- **Conclusion**: There are **no solutions** for \( x \) and \( y \) when \( c < 0 \).

### Case 2: \( c = 0 \)

For the equation \( ax^2 + by^2 = c \), when \( c = 0 \):

- **Justification**: \( ax^2 + by^2 = 0 \) implies that both \( ax^2 = 0 \) and \( by^2 = 0 \). This can only happen if \( x = 0 \) and \( y = 0 \).
- **Conclusion**: There is exactly **one solution** for \( x \) and \( y \), which is \( (x, y) = (0, 0) \).

### Case 3: \( c > 0 \)

For the equation \( ax^2 + by^2 = c \), when \( c > 0 \):

- **Justification**: Since \( ax^2 + by^2 = c \) and \( c > 0 \), there could be various pairs of \( (x, y) \) that satisfy the equation, depending on the values of \( a \) and \( b \). Generally, this equation represents an ellipse (or a circle if \( a = b \)) when \( a > 0 \) and \( b > 0 \), so there will be infinitely many pairs \( (x, y) \) that satisfy this equation.
- **Conclusion**: There are **infinitely many solutions** for \( x \) and \( y \) when \( c > 0 \).

### Summary
1. **Case 1** (\( c < 0 \)): No solutions.
2. **Case 2** (\( c = 0 \)): Exactly one solution, \( (0, 0) \).
3. **Case 3** (\( c > 0 \)): Infinitely many solutions.

Do you want details or have any questions? Here are some related questions you might find interesting:

1. How do the values of \( a \) and \( b \) affect the number of solutions in the given equation?
2. What if the equation was \( ax^2 + by^2 + c = 0 \)? How would the number of solutions change?
3. How does the concept of a discriminant apply when analyzing the number of solutions in a quadratic equation?
4. Can we determine the number of solutions for a linear system of equations in the same manner?
5. What happens if the equation is non-homogeneous, i.e., of the form \( ax^2 + by^2 + dx + ey + f = 0 \)?

**Tip**: When solving equations, always check the nature of the equation (linear, quadratic, etc.) as it helps in determining the possible number of solutions effectively.

Understand how to determine the number of solutions for quadratic equations with variable coefficients (x and y) in different scenarios (c < 0, c = 0, c > 0). Detailed explanations provided.

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