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Question

If the system of linear equations
x+y+z=5
x+2y+3z=9
x+3y+αz=β
has infinitely many solutions, then βα equals:

A
5
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B
18
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C
8
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D
21
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Solution

The correct option is C 8
Δ=∣ ∣11112313α∣ ∣
=1(2α9)1(α3)+1(32)
=2α9α+3+1
=α5
For infinitely many solutions,
Δ=0
α5=0
α=5

Now,
Δ1=∣ ∣11512913β∣ ∣
=1(2β27)1(β9)+5(32)
=2β27β+9+5
=β13
For infinitely many solutions,
Δ1=0
β13=0
β=13

Hence, βα=135=8

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