Question

For the pair of equations $\lambda$ x+3y+7=0 and 2x + 6y – 14 = 0 to have infinitely many solutions, the value of $\lambda$ should be 1. Is the statement true? Give reasons

Answer 1

The given pair of linear equations are:
$\lambda x+3y+7=0$ and $2x+6y–14=0$

Here, $a_1=\lambda ,b_1=3,c_1=7a_2=2,b_2=6,c_2=-14$

$If\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$, then the pair of equations has infinitely many solutions

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

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

$\Rightarrow \lambda =1.....\left(i\right)$

Also, $\frac{\lambda }{2}=-\frac{7}{14}$

$\Rightarrow \lambda =-1$

hence, $\lambda$ does not have a unique value for the given condition.
Or, we can say that for no value of $\lambda$ will the given pair of linear equations have infinitely many solutions.

Thus, the given statement is not true.