The value of λfor which the following system of equation...

The value of $\lambda$ for which the following system of equation does not have a solution is
$$
\begin{array}{l}
x+y+z=6 \\
4 x+\lambda y-\lambda z=0 \\
3 x+2 y-4 z=-8
\end{array}
$$
(a) 3
(b) $-3$
(c) 0
(d) 1

Key Concepts

System of Linear Equations

A system of linear equations involves multiple linear equations that share the same variables. The overall goal is to find values for these variables that satisfy all the equations simultaneously. The nature of the solutions (unique, infinitely many, or none) depends on the relationships between the equations.

Parameter-Dependent Systems

Parameter-dependent systems include one or more parameters whose values can affect the behavior of the system. Analyzing how these parameters influence the consistency and the solution set of the system is crucial, as certain values may cause the system to become inconsistent or change the number of solutions.

Inconsistency in Linear Systems

Inconsistency in a system of linear equations occurs when no possible set of variable values can satisfy all the equations at the same time. This situation is often detected when an equation simplifies to a contradiction (for example, a statement like 0 = nonzero), typically identified by comparing the ranks of the coefficient matrix and the augmented matrix.

Elimination Method

The elimination method is a technique used to solve systems of linear equations by successively eliminating variables. By substituting or combining equations, the system is reduced step-by-step, which helps identify conditions that lead to a unique solution, infinitely many solutions, or an inconsistency in the system.

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