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Question

Consider the following system of linear equations 
2144312128xyz=a57
Number of values of a for which system has infinitely many solutions is 


Solution

For the system 2144312128xyz=a57
has infinitely many solutions 

D=∣ ∣2144312128∣ ∣=0

D1=0, for all values of a as C2 and C3 are identical.

D2=0∣ ∣2a44512178∣ ∣=0

Applying R2R23R1, R2R32R1
∣ ∣2a4253a0372a0∣ ∣=0a=15

Now, D3=0
∣ ∣21a435127∣ ∣=0

Applying R2R22R1, R3R30.5R1
∣ ∣21a0152a01.570.5a∣ ∣=0a=15

Hence, there is only one value of a for which the system has infinitetly many solution.
 

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