The values of $k$ , for the following system of linear equations which has an

infinite number of solutions $x+\left(k+1\right)y=5,\left(k+1\right)x+9y=8k-1$ is

Given equations $x+\left(k+1\right)y=5,\left(k+1\right)x+9y=8k-1$

Here

$\begin{array}{c}{a}_1=1,b_1=\left(k+1\right),c_1=-5,\\ a_2=k+1,b_2=9,c_2=-\left(k-1\right)\end{array}$

Given equation has infinite solution

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

$\frac{1}{k+1}=\frac{k+1}{9}=\frac{-5}{-\left(8k-1\right)}$

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

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

$k^2+1+2k-9=0$

$k^2+2k-8=0$

$k^2+4k-2k-8=0$

$k\left(k+4\right)-2\left(k+4\right)=0$

$\left(k+4\right)=0;k-2=0$

$k=-4\left(or\right)k=2$

If $k=2$  given system of equations has infinite number of solutions.