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Question

If the system of equations

x+y+z=22x+4yz=63x+2y+λz=μ

has infinitely many solutions, then


A

λ-2μ=-5

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B

2λ+μ=14

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C

λ+2μ=14

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D

2λ-μ=5

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Solution

The correct option is B

2λ+μ=14


Explanation for the correct option.

Step 1. Find the value of λ.

The given system has infinitely many solutions so D=D1=D2=D3=0.

Now for the system x+y+z=22x+4yz=63x+2y+λz=μ taking D=0 the equation can be formed as:

11124-132λ=014λ+2-12λ+3+14-12=04λ+2-2λ-3-8=02λ=9λ=92

Step 2. Find the value of μ.

Again solve for D3=0.

11224632μ=014μ-12-12μ-18+24-12=04μ-12-2μ+18-16=02μ=10μ=5

Step 3. Find the relation.

For λ=92 and μ=5, the value of 2λ+μ is:

2λ+μ=2×92+5=9+5=14

Hence, the correct option is B.


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