Determinants · Question 32

If the system of equations 2x + y + 3z = 5 3x + 2y - z = 7 4x + 5y + z = 9 has infinitely many solutions, then ( 2 + 2) is equal to :

1. 30
2. 26
3. 22
4. 18

Answer

(2) 26

Solution

The 2nd equation is not a multiple of the 1st (x and z coefficients 3, −1 against 2, 3), so for infinitely many solutions the 3rd equation must be a combination of the first two:

(4x + 5y + z - 9) = p(2x + y + 3z - 5) + q(3x + 2y - z - 7)

x: 2p + 3q = 4

constant: 5p + 7q = 9

p = -1, q = 2

y: p + 2q = 5 - + 4 = 5 = -1

z: 3p - q = = -3 - 2 = -5

Check: with = -1, = -5, = vmatrix 2 & -1 & 3 \ 3 & 2 & -1 \ 4 & 5 & -5 vmatrix = 2(-5) + 1(-11) + 3(7) = 0

2 + 2 = 1 + 25 = 26

Answer: (2)