Question

For what value of k, will the following system of equations has infinitely many solutions?
$2x+3y=4$
$(k+2)x+6y=3k+2$

Answer 1

We know that the system of equations
$a_{1}x+b_{1}y=c_1$
$a_{2}x+b_{2}y=c_2$

has infinitely many solutions, if
$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

Here, $a_1=2,b_1=3,c_1=4,a_2=k+2,b_2=6,c_2=3k+2$.

Therefore, the given system of equations will have infinitely many solutions, if

$\frac{2}{k+2}=\frac{3}{6}=\frac{4}{3k+2}$

$\Rightarrow \frac{2}{k+2}=\frac{3}{6}$ and $\frac{3}{6}=\frac{4}{3k+2}$

$\Rightarrow \frac{2}{k+2}=\frac{1}{2}$ and $\frac{1}{2}=\frac{4}{3k+2}$

$\Rightarrow k+2=4$ and $3k+2=8$

$\Rightarrow k=2$ and $k=2$

$\Rightarrow k=2$

Hence, the given system of equations will have infinitely many solutions, if $k=2$.