System Of Linear Equations

We are given the system of equations:

We can write this system in terms of a family of planes. Using the family of planes, we have:

Expanding and simplifying:

Solving this system, we find:

Now, calculate :

Concept: For a system of three linear equations:

• If → unique solution.
• If and the augmented matrix has different rank → no solution.
• If and ranks are equal → infinitely many solutions.

Thus, for the system to have no solution, the determinant of the coefficient matrix must be zero but the equations must remain inconsistent.
Step 1:Form the coefficient matrix. For no solution,
Step 2:Compute the determinant.} Expanding along the first row: For no solution,
Step 3:Check the consistency condition. Multiply the first equation by : Multiply the second equation by : The third equation is Substituting , the left-hand sides become proportional, but the constants are not proportional, hence the system becomes inconsistent. Thus,

For the system to have infinitely many solutions, the coefficient matrix must be singular, meaning its determinant should be zero. The coefficient matrix is: The determinant of this matrix should be zero for infinitely many solutions. Let's calculate the determinant: Setting the determinant equal to zero: Now substitute into the third equation to find the relationship between and . Simplifying gives the line equation , so the correct answer is .
Final Answer: (C)

Step 1: Understanding the Concept:
For a system to have infinitely many solutions, the determinant of the coefficient matrix ( ) must be zero, and the constants must be consistent with the linear combination of the other equations.

Step 2: Key Formula or Approach:
1. Set .
2. Use row transformations to check consistency or set .

Step 3: Detailed Explanation:
.
:
.
For infinite solutions, the constant terms must follow the same row relationship.
Performing row operations on the augmented matrix :
.
.
For these to be identical: and .
.

Step 4: Final Answer:
The value is 18.