Find a number $t$ such that

Is not linearly independent in $\mathbf R^3$

We want (for some $a_k \ne 0$)

If some $a_k \ne 0$, we must have $a_3 \ne 0$ since $(3,1,4)$ and $(2,-3,5)$ are linearly independent. Therefor we can divide by $a_3$, relabel and rearrange to get

We'll pick $b_1, b_2$ to satisfy the first two constraints (the first coordinates must be equal) then pick $t$ to satisfy the last one.

Which gives $t = 3(4) - 2(5) = 2$, which is our solution since

Implying they are linearly dependent.

(I could have skipped the derivation and jumped to this part, but that makes it harder to understand)