If the system fo equations
x+y+z = 5
x + 2y + 3z = 9
`x + 3y + alphaz = beta`
has infinitely many solution, then `beta - alpha` equals

Options

A. 5
B. 21
C. 18
D. 8

Answer

D

Explanation

`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`