System of linear equation have no solution,

$$\therefore\,\,\,$$ determinant of coefficient = 0

$$\left| {\matrix{ {k + 2} & {10} \cr k & {k + 3} \cr } } \right| = 0$$

$$ \Rightarrow $$ $$\,\,\,\,$$ (k + 2) (k + 3) $$-$$ 10 K = 0

$$ \Rightarrow $$ $$\,\,\,\,$$ k² $$-$$ 5k + 6 = 0

$$\therefore\,\,\,\,$$ k = 2, 3

When, k = 2 then equations become,

4x + 10y = 2

and 2x + 5y = 1

It has in finite number of solutions.

When k = 3, equations becomes

5x + 10y = 3

3x + 6y = 2

Those equation has no solutions.

$$\therefore\,\,\,\,$$ When k = 3, then system of equations have no solutions.