If the system of equations $2 x+7 y+\lambda z=3$ $3 x+2 y+5 z=4$ $x+\mu y+32 z=-1$ has infinitely many solutions, then $(\lambda-\mu)$ is equal to ______.

JEE Main · 2024 · 06 Apr 2024 · Shift 2 · Determinants

If the system of equations

2 x + 7 y + λ z = 3

3 x + 2 y + 5 z = 4

x + μ y + 32 z = - 1

has infinitely many solutions, then

(λ - μ)

is equal to ______.

Answer: 38

For the system to have infinitely many solutions, the third equation must be a linear combination of the first two:

x + μ y + 32 z = a (2 x + 7 y + λ z) + b (3 x + 2 y + 5 z)

, matching the right-hand sides too:

- 1 = 3 a + 4 b

. Solving

2 a + 3 b = 1

and

3 a + 4 b = - 1

gives

a = - 7, b = 5

. Then

μ = 7 a + 2 b = - 49 + 10 = - 39

, and

32 = λa + 5 b = - 7 λ + 25 \Rightarrow λ = - 1

. Hence

λ - μ = - 1 - (- 39) = 38

JEE Main · 2024 · 06 Apr 2024 · Shift 2 · Integer answer