Above is the question. So far I have calculated that the orthogonal vectors are

v1'=(1,2,3,-1,2); v2'=((11/19),(22/19),(52/19),(65/19),(-73/19))

but how do I make a system of linear equations having these vectors as solutions?

I understand how to make a system using only one vector:

x1 = 1

x2 = 2

x3 = 3

x4 = -1

x5 = 2

but I'm unsure how to have both orthogonal vectors as solution to the system.