Assertion :Consider the system of equations, $2 x + 3 y + 4 z = 5$ , $x + y + z = 1$ , $x + 2 y + 3 z = 4$ This system of equations has infinite solutions. Reason: If the system of equations is $e_{1} : a_{1} x + b_{1} y + c_{1} z - d_{1} = 0$ , $e_{2} : a_{2} x + b_{2} y + c_{2} z - d_{2} = 0$ , $e_{3} : e_{1} + λ e_{2} = 0$ , where $λ ϵ R$ & $\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$ Then such system of equations has infinite solutions.

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

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Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

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Assertion is correct but Reason is incorrect

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Assertion is incorrect but Reason is correct

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Solution

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
Reason is correct.
Given system of equations
$2 x + 3 y + 4 z = 5$ ....(1)
$x + y + z = 1$ ....(2)
$x + 2 y + 3 z = 4$ ....(3)
Adding (2) and (3), we get
$2 x + 3 y + 4 z = 5$
For eqn (2) and (3), we have
$\frac{a_{1}}{a_{2}} = 1$ , $\frac{b_{1}}{b_{2}} = \frac{1}{2}$
$\Rightarrow \frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$
Hence, given system of equation has infinitely many solutions. Reason is the correct explanation for assertion.

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