If the system of linear equations
x+ y +z = 5
x+2y +2z = 6
`x + 3y + lambdaz = mu, (lambda, mu in R)` has infinitely many solutions, then the value of `lambda + mu` is

Options

A. 7
B. 12
C. 10
D. 9

Explanation

Given system of linear equations
`x + y +z = 5 " "…(i)`
`x + 2y + 2z = 6 " " …(ii)`
`x+ 3y + lambdaz = mu " " …(iii)`
`(lambda, mu in R)`
The above given system has infinitely many solutions, then the plane represented by these equations
intersect each other at a line, means `(x+3y + lambdaz -mu)`
` = p(x + y + z -5) +q (x + 2y +2z-6)`
`= (p +q)x + (p +2q)y + (p + 2q)z - (5p + 6q)`
On comparing, we get
`p +q =1, p + 2q = 3, p+2q = lambda`
`"and " 5p + 6q = mu`
So, (p, q) = (-1, 2)
`rArr lambda = 3 " and " mu =7`
`rArr lambda + mu = 3 + 7 = 10`