You can try SolveValues
instead of Solve
:
sys = {-x1 + y1 + y5 == 0,
-x4 + y2 == 0,
y3 - 2 x5 y4 == 0,
-1 + 2 x4 + y4 == 0,
-x2 + x5 + y5 == 0};
al = {y1, y2, y3, y4, y5, x1, x2, x3, x4, x5};
SolveValues[sys, al]
Thread[al -> %[[1]]]
sys /. % // Simplify