  In this paper, we give elementary proofs of Zagier's formula for multiple
zeta values involving Hoffman element and its odd variant due to Murakami.
Zagier's formula was a key ingredient in the proof of Hoffman's conjecture.
Moreover, using the same approach, we prove Murakami's formula for multiple
$t$-values. This formula is essential in proving a Brown type result which
asserts that each multiple zeta value is a $\mathbb{Q}$-linear combination of
multiple $t$-values of the same weight involving $2$'s and $3$'s.
