For the pair of equations λx + 3y + 7 = 0
Question:

For the pair of equations λx + 3y + 7 = 0 and 2x + 6y -14 = 0. To have infinitely many solutions, the value of λ should be 1. Is the statement true? Give reasons.

Solution:

No, the given pair of linear equations

λx + 3y+7 = 0 and 2x + 6y-14 = 0

Here,                         a1= λ, b1 = 3 c1 =7; a2 =2, b2 = 6,c2 = -14

If $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$, then system has infinitely many solutions.

$\Rightarrow$ $\frac{\lambda}{2}=\frac{3}{6}=-\frac{7}{14}$

$\because$ $\frac{\lambda}{2}=\frac{3}{6} \Rightarrow \lambda=1$

and $\frac{\lambda}{2}=-\frac{7}{14} \Rightarrow \lambda=-1$

Hence, λ = -1 does not have a unique value.                                                     ‘

So, for no value of λ the given pair of linear equations has infinitely many solutions.