For what value of k will the following system of equations have infinitely many solutions?

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I've been struggling for hours trying to solve this: For what value of k does the following system of linear equations have infinitely many solutions?

$$x+y+kz=3$$ $$x+ky+z=-7$$ $$kx+y+z=4$$

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$2x+3y=4and(k+2)x+6y=3k+2$

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

$\Rightarrow 6k+4=4k+8$

$\Rightarrow$ $k=2$

Role of Coefficients and Constants