When One Equation Says Nothing New

A system of two equations in $x$ and $y$ carries an unknown constant $k$:

$(k+5)\,x + 6y = k+7$

$4x + ky = 5$

For almost every value of $k$ this system has exactly one solution. There are precisely two exceptional values of $k$: at one of them the system has no solution at all, and at the other it has infinitely many.

Identify the value of $k$ that produces infinitely many solutions. At that $k$, every solution of the system lies on one single straight line.

On that line, what is the value of $y$ when $x = -2$?

Give your answer rounded to two decimal places. The answer is a plain number with no units.