Prove a feasible point is optimal for an LP using complementary slackness

You are right so far. Here you have the case with a multiple optimal solution:

$$(y_1^*,y_2^*)=\left(y_1,1-\frac72y_1\right)$$

with the constraint $1-\frac72y_1\leq 0$. It comes out that $y_1\geq \frac27$.

So one possible optimal solution is $(y_1^*,y_2^*)=\left(\frac47,-1\right)$

Remark

Your dual is almost right. If the variables of a min primal problem are non-negative, then the corresponding constraints are $\leq$-constraints. So the $t_j$'s are added and not subtracted.

Maximize $14y_1+4y_2$
$7y_1+2y_2+t_1=2$
$6y_1+4y_2+t_2=5$
$3y_1+5y_2+t_3=7$
Where: $y_1,t_1,t_2,t_3 \ge 0,y_2 \le 0$