https://file.aiquickdraw.com/mathbot/file/1728691162232h7frj2rf.jpg

The system of equations provided in the image is as follows:

\[
\begin{aligned}
1. & \quad x + y = 0 \\
2. & \quad y + z = 0 \\
3. & \quad x + z = 0 \\
4. & \quad ax + by + cz = 0
\end{aligned}
\]

The accompanying instructions ask for three things:
1. **Find a unique solution for \( (a, b, c) \)**.
2. **Find a case where there is no solution for \( (a, b, c) \)**.
3. **Find values for \( (a, b, c) \) that would result in an infinite number of solutions**.

### Step-by-Step Approach:

#### 1. **Solving the first three equations:**

From the first three equations, we can simplify the system:

\[
x + y = 0 \quad \Rightarrow \quad x = -y
\]
\[
y + z = 0 \quad \Rightarrow \quad z = -y
\]
\[
x + z = 0 \quad \Rightarrow \quad -y + (-y) = 0 \quad \Rightarrow \quad -2y = 0 \quad \Rightarrow \quad y = 0
\]

Thus, we find:
\[
y = 0, \quad x = 0, \quad z = 0
\]

The solution to the first three equations is \( x = y = z = 0 \).

#### 2. **Unique Solution for \( (a, b, c) \):**

For a unique solution, we require that the fourth equation \( ax + by + cz = 0 \) is consistent with the first three equations, which yield \( x = 0, y = 0, z = 0 \). Hence, the fourth equation simplifies to:
\[
a(0) + b(0) + c(0) = 0 \quad \Rightarrow \quad 0 = 0
\]
This equation is always true, regardless of the values of \( a, b, c \). Therefore, any values of \( (a, b, c) \) will give a unique solution, which is \( x = y = z = 0 \).

#### 3. **No Solution for \( (a, b, c) \):**

To have no solution, the fourth equation must contradict the zero solution from the first three equations. For example, if the fourth equation is something like:
\[
ax + by + cz = k
\]
where \( k \neq 0 \), then the equation becomes inconsistent because it simplifies to \( 0 = k \), which is a contradiction.

Thus, a specific case with no solution could be:
\[
ax + by + cz = 1
\]
where \( a, b, c \) can take any non-zero values, as long as the right-hand side is non-zero.

#### 4. **Infinite Solutions for \( (a, b, c) \):**

For infinite solutions, the fourth equation must be dependent on the first three, meaning that it does not introduce any new constraints. One way to achieve this is if the fourth equation is a linear combination of the first three. Since the first three equations lead to \( x = y = z = 0 \), the fourth equation must essentially be redundant.

For example, if \( a = 1 \), \( b = 1 \), and \( c = 1 \), then the fourth equation becomes:
\[
x + y + z = 0
\]
which is already satisfied by the first three equations. Thus, in this case, the system has infinitely many solutions because the fourth equation does not introduce any new information.

### Summary:
1. **Unique solution**: Any values of \( a, b, c \) give the unique solution \( x = y = z = 0 \).
2. **No solution**: For example, if \( ax + by + cz = 1 \), then there is no solution.
3. **Infinite solutions**: A case could be when \( a = 1 \), \( b = 1 \), and \( c = 1 \), making the fourth equation redundant.

Would you like to explore these solutions further or have any specific questions? 

Here are five related questions to deepen your understanding:
1. What are the conditions for a system of linear equations to have a unique solution?
2. How can the determinant of a coefficient matrix help in determining the number of solutions?
3. Can you describe the role of linear dependence and independence in systems of equations?
4. How can we use matrix representation to generalize the solution for such systems?
5. What happens to the solution when parameters \( a, b, c \) are varied systematically?

**Tip**: A system of linear equations has no solution when the equations are inconsistent, meaning they represent parallel planes that do not intersect.

The system of equations provided is: x + y = 0, y + z = 0, x + z = 0, ax + by + cz = 0. The task is to find a unique solution, no solution, and an infinite solution for (a, b, c).

Explore the solutions to a system of linear equations with variables x, y, z and parameters (a, b, c). Learn how to find unique, no solution, and infinite solution cases using algebraic methods. Suitable for high school students (Grades 10-12).

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