it's probably just an arithmetic error, all your work is correct.

when we add the linear combinations we get:

((1+t)/2)(1,2) + ((1-3t)/2)(5,6) + t(7,8) = (((1+t)/2 + (5-15t)/2 + 7t), (1+t + 3-9t + 8t))

= ((6-14t)/2 + 7t, 4-8t + 8t) = (3 - 7t + 7t, 4 - 8t + 8t) = (3,4)