Determinants
General
Grade 12
Question:
Find the value of $\lambda$, if the following equations are consistent: $x + y - 3 = 0$; $(1 + \lambda)x + (2 + \lambda)y - 8 = 0$; $x - (1 + \lambda)y + (2 + \lambda) = 0$
Step-by-Step Solution
Key Concept: General
The given equations in two unknowns are consistent, then $\Delta = 0$<br>i.e. $\begin{vmatrix} 1 & 1 & -3 \\ 1+\lambda & 2+\lambda & -8 \\ 1 & -(1+\lambda) & 2+\lambda \end{vmatrix} = 0$<br>Applying $C_2 \rightarrow C_2 - C_1$ and $C_3 \rightarrow C_3 + 3C_1$<br>$\therefore \begin{vmatrix} 1 & 0 & 0 \\ 1+\lambda & 1 & 3\lambda-5 \\ 1 & -2-\lambda & 5+\lambda \end{vmatrix} = 0$<br>$\Rightarrow (5+\lambda) - (3\lambda-5)(-2-\lambda) = 0 \Rightarrow 3\lambda^2 + 2\lambda - 5 = 0$<br>$\therefore \lambda = 1, -5/3$
Correct Answer: 1, -5/3