If the system of equations x+4 y-z=λ, 7 x+9 y+μ z=-3,5 x+y+2 z=-1 has infinitely many solutions, then (2 μ+3 λ) is equal to :

Solution

Maths sir · 4 steps

Infinitely many solutions means one equation is a combination of the other two, so find that combination.

1. Determinant first. For infinitely many solutions the coefficient determinant must be zero. Expanding along the first row gives D = 1(18-μ) - 4(14-5μ) - 1(7-45) = 19μ, so μ = 0.
2. Find the combination. Look for a and b with (first equation)× a + (third equation)× b = second equation. Matching x gives a + 5b = 7 and matching y gives 4a + b = 9, so b = 1 and a = 2.

   2(x+4y-z) + (5x+y+2z) = 7x + 9y + 0· z = 2λ - 1

3. Match the rest. The z-coefficient is -2 + 2 = 0, which agrees with μ = 0. The right side must match too: 2λ - 1 = -3, so λ = -1.
4. So. 2μ + 3λ = 2(0) + 3(-1) = -3, which is option (d).

The idea to keep. Infinitely many solutions needs D = 0 and also consistent right sides. Writing one equation as a combination of the other two checks both at once.

Answer from the NTA final answer key. Solution by PrepWiser.