For the pair of equations
$ \lambda x+3 y=-7 $
$ 2 x+6 y=14 $
to have infinitely many solutions, the value of $ \lambda $ should be 1 . Is the statement true? Give reasons.

Given :

The given pair of equations is,

\( \lambda x+3 y=-7 \)
\( 2 x+6 y=14 \)

To find :

We have to find whether the value of \( \lambda \) is 1.

Solution:

We know that,

The condition for infinitely many solutions is,

$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$            

\( \lambda x+3 y+7=0 \)
\( 2 x+6 y-14=0 \)

Here,

$a_1=\lambda, b_1=3, c_1=7$

$a_2=2, b_2=6, c_2=-14$

Therefore,

$\frac{a_1}{a_2}=\frac{\lambda}{2}$

$\frac{b_1}{b_2}=\frac{3}{6}=\frac{1}{2}$

$\frac{c_1}{c_2}=\frac{7}{-14}=-\frac{1}{2}$

This implies,

$\frac{\lambda}{2}=\frac{1}{2}$

$\lambda=1$

$\frac{\lambda}{2}=-\frac{1}{2}$

$\lambda=-1$

Here,

$\lambda$ does not have a unique value.

Hence, for no value of $\lambda$ the given pair of linear equations has infinitely many solutions.

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Updated on: 10-Oct-2022

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