If the system of equations $2x + 5y = 7, 6x + \lambda y = 28$ is inconsistent, then

An inconsistent system means the equations have NO solution — the two lines are parallel and never meet.

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For the system

$2x + 5y = 7$

$6x + \lambda y = 28$

the condition for inconsistency is:

$\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}$

This means the lines have the same slope (parallel) but different intercepts (never intersect).

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Here, $a_1 = 2,\; a_2 = 6,\; b_1 = 5,\; b_2 = \lambda,\; c_1 = 7,\; c_2 = 28$

Applying the first part of the condition:

$\dfrac{2}{6} = \dfrac{5}{\lambda}$

$\dfrac{1}{3} = \dfrac{5}{\lambda}$

Cross-multiplying:

$\lambda = 5 \times 3 = 15$

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Now we verify the "not equal to" part to confirm the system is truly inconsistent and not dependent:

$\dfrac{a_1}{a_2} = \dfrac{1}{3}$

$\dfrac{c_1}{c_2} = \dfrac{7}{28} = \dfrac{1}{4}$

$\dfrac{1}{3} \neq \dfrac{1}{4}$ ✅

This confirms the system is inconsistent.

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$\lambda = 15$