Find the value of k for which each of the following system of equations have infinitely many solutions :

GIVEN:

$x+(k+1) y=4$

$(k+1) x+9 y=5 k+2$

To find: To determine for what value of k the system of equation has infinitely many solutions 

We know that the system of equations

$a_{1} x+b_{1} y=c_{1}$

$a_{2} x+b_{2} y=c_{2}$

For infinitely many solution 

$\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$

Here,

$\frac{1}{k+1}=\frac{(k+1)}{9}=\frac{4}{5 k+2}$

$\frac{1}{k+1}=\frac{(k+1)}{9}$

$9=(k+1)^{2}$

$3^{2}=(k+1)^{2}$

$k+1=3$

$k=2$

Hence for $k=2$ the system of equation have infinitely many solutions.