First slide

System of linear equations

Question

$If the system of equations x + y + z = 0, x + 2 y + 3 z = 1, λ x + μ y + 4 z = 1 has infinitely many solutions then (λ, μ) =$

Moderate

A

(2, 3)
B

(3, 2)
C

(1, 2)
D

(2, 1)

Solution

$\begin{array}{l} x + y + z = 0, \dots \dots \dots (1) \\ x + 2 y + 3 z = 1 \dots \dots \dots (2) \\ λ x + μ y + 4 z = 1 \dots \dots (3) \\ 3 \times (1) - (2) gives 2 x + y = - 1 \dots \dots \dots . . . (4) \\ 4 \times (1) - (3) gives (4 - λ) x + (4 - μ) y = - 1 \dots (5) \\ (4), (5) have infinitely many solutions \\ \Rightarrow \frac{2}{4 - λ} = \frac{1}{4 - μ} = 1 \\ \Rightarrow λ = 2, μ = 3 \end{array}$

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