Question

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. 8
B. 18
C. 21
D. 5

Answer 1

Since, the system of equations has infinitely many solution, therefore `D = D_(1) = D_(2) =D_(3)=0`
Here,
`D= |{:(1, 1, 1), (1, 2, 3), (1, 2, alpha):}| = 1(2alpha -9)-1 (alpha-3) + 1(3-2)`
` = alpha -5`
`"and "D_(3) = |{:(1, 1, 5), (1, 2, 9), (1, 3, beta):}| = 1(2beta -27) -1(beta -9) + 5(3-2)`
`= beta-13`
Now, D = 0
`rArr alpha -5 =0 rArr alpha = 5`
`"and " D_(3) =0 rArr beta -13 =0`
`rArr beta = 13`
`therefore beta -alpha = 13-5 =8`