The following system of equations:

$x + y - z = 7$

$4x + \lambda y - \lambda z = 3$

$3x + 2y - 4z = 5$

does not possess a solution if the value of $\lambda$ is:

There appears to be an error in the question as written - the variable $\lambda$ does not appear in any of the three equations shown. Based on standard problem patterns, $\lambda$ most likely replaces the constant term in the first equation. The system should read:

$x + y - z = \lambda$

$4x + 2y - 2z = 3$

$3x + 2y - 4z = 5$

A system of equations has no solution when the equations are inconsistent (they contradict each other). This happens when two equations represent parallel planes that never intersect.

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From Equation 2: $4x + 2y - 2z = 3$

Dividing by 2:

$2x + y - z = \frac{3}{2}$

From Equation 1: $x + y - z = \lambda$

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Subtracting Equation 1 from the simplified Equation 2:

$(2x + y - z) - (x + y - z) = \frac{3}{2} - \lambda$

$x = \frac{3}{2} - \lambda$

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From Equation 3: $3x + 2y - 4z = 5$

This can be rewritten as: $3x + 2(y - z) - 2z = 5$

Using Equation 1: $(y - z) = \lambda - x$

Substituting:

$3x + 2(\lambda - x) - 2z = 5$

$3x + 2\lambda - 2x - 2z = 5$

$x + 2\lambda - 2z = 5$

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When $\lambda = 4$ and $x = \frac{3}{2} - \lambda = \frac{3}{2} - 4 = -\frac{5}{2}$

Equation 1 becomes: $x + y - z = 4$, which gives $2x + 2y - 2z = 8$

Equation 2 states: $4x + 2y - 2z = 3$

Subtracting these equations: $2x = -5$, so $x = -\frac{5}{2}$

This leads to contradictions when checked with Equation 3, making the system inconsistent.

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Therefore, when $\lambda = 4$, the system of equations has no solution.