Example 1 - mental math
It can be observed that the value of b in the bottom equation is five times of the top equation. For the system to have one solution, the value of a = k in the bottom equation cannot be 5 times the value of a in the top equation. Hence, k cannot be 10.
Example 3 - shortcut and may be mental math
The question asks for the value of 50x + 10y. It can be observed that if the two equations are added, the result is the value of 5x + y. The value can be multiplied by 10 to obtain the answer.