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If the system of equations x+ y+z = 5...

If the system of equations
x+ y+z = 5 x+ 2y + 3z =9
x +3y + az = `beta`
has infinitely many solutions, then `beta-alpha` equals

A

5

B

21

C

18

D

8

Text Solution

Verified by Experts

The correct Answer is:
D
`Delta=abs({:(1,1,1),(1,2,3),(1,3,alpha):})`=0
`2alpha-9+3-alpha+3-2=0`
`alpha-5=0`
`alpha=5`
also `abs({:(1,1,5),(2,3,9),(3,alpha,beta):})`=0
`abs({:(1,1,5),(2,3,9),(3,5,beta):})`=0
`3beta-45+27-2beta+5=0`
`beta-13=0`
`beta=13`
`beta-alpha=8`
OR
Since the system has infinitely many situations, therefore the system of equations must be linearly dependent. Not hard to see
`2(B)-(A)=CrArralpha=5,beta=13`
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