8. What should be the values of \( a \) and \( b \) so that the system \( \left\{\begin{array}{l}x+y+z=0 \\ 3 x-a y-z=4 \\ x+5 y+5 z=b\end{array}\right. \) has no solution.

To determine the values of \( a \) and \( b \) for which the system has no solution, we need to consider the conditions under which a system of linear equations is inconsistent. Start by rewriting this system in matrix form and performing row operations. We expect that for the system to have no solution, the equations must represent planes that do not intersect at any point. A common situation arises when the third equation can be expressed as a linear combination of the first two equations, leading to a contradiction. For instance, if \( a = 3 \) and \( b = 4 \), the second and third equations become dependent, while the first adds a dimension, causing inconsistency in solution paths. Hence, this choice introduces the condition that will yield no solution. So, set \( a = 3 \) and \( b = 4 \) to ensure that the system has no solutions!