Let
Y1 = (x + 37)/5
Y2 = (7x - 1)/2

"Two consecutive even integers" means that (Y1 + Y2)/2 = Y1 + 1

Therefore, [(x + 37)/5 + (7x - 1)/2]/2 = [(x + 37)/5] + 1

Solve for x. OP blew the original problem by leaving out parentheses. As stated, there was no solution.