The system of linear equations given is:

1. $x - 3y - 10z = -1$
2. $-2x + 5y + 18z = b$
3. $x - 5y + bz = 3a$

To determine the conditions under which the system will have infinitely many solutions, no solutions, or a unique solution, we need to analyze the system using the concepts from linear algebra (determinants, rank, etc.).

Infinitely Many Solutions:

A system of linear equations will have infinitely many solutions if the system is dependent and consistent. This occurs if the determinant of the coefficient matrix is zero, and the equations describe the same geometric object (like planes in space intersecting along a line).

In order to find the conditions on $a$ and $b$ for infinitely many solutions, we will need to analyze the augmented matrix and check when it becomes consistent and dependent.

No Solution:

The system will have no solution if the equations are inconsistent. This happens when the coefficient matrix is singular (determinant is zero), but the augmented matrix represents conflicting equations (e.g., parallel planes that never intersect).

Unique Solution:

The system will have a unique solution if the determinant of the coefficient matrix is non-zero. This implies that the system is independent and consistent, meaning the three planes intersect at exactly one point.

I'll now begin solving and finding the conditions for $a$ and $b$.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you'd like, I can explain the approach further or help you solve it manually step by step. Would that work for you?

Here are some related questions to think about:

1. How do you find the determinant of a 3x3 matrix?
2. What is the significance of a zero determinant in a system of equations?
3. How do you check for consistency in a system of linear equations?
4. What does it mean for a system to be dependent or independent?
5. How can augmented matrices help in determining solution types?

Tip: The determinant of a matrix reveals whether the system is solvable (non-zero) or if the equations are dependent (zero determinant).