x - 3y - 10z = -1 , -2x + 5y +18z = b, x - 5y +bz = 3a    will have infinitely many solutions if 𝑏Answer 1 Question 15Answer 2 Question 15 and 𝑎Answer 3 Question 15Answer 4 Question 15,

it will have no solution if 𝑏Answer 5 Question 15Answer 6 Question 15 and 𝑎Answer 7 Question 15Answer 8 Question 15,

and it will have a unique solution if 𝑏Answer 9 Question 15Answer 10 Question 15

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).

The query submitted by the user: x - 3y - 10z = -1 , -2x + 5y + 18z = b, x - 5y + bz = 3a will have infinitely many solutions if bAnswer 1 Question 15Answer 2 Question 15 and aAnswer 3 Question 15Answer 4 Question 15, it will have no solution if bAnswer 5 Question 15Answer 6 Question 15 and aAnswer 7 Question 15Answer 8 Question 15, and it will have a unique solution if bAnswer 9 Question 15Answer 10 Question 15.

Explore the conditions under which a system of linear equations involving parameters 'a' and 'b' will have infinitely many solutions, no solutions, or a unique solution. This problem touches on key linear algebra concepts such as determinants, system consistency, and dependence. Suitable for advanced high school or college students.

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