Answer: Yes, there is an ordered pair solution
The quick way we can see this without having to graph or do any algebra is to look at the two slopes of the lines.
y = x+1 has slope of 1
y = -x+5 has slope of -1
Recall that y = mx+b has slope m.
The two different slopes (1 and -1) indicate the lines are not parallel, so they must intersect exactly once. This intersection is the where the solution is located.