Let F be a free algebra in a variety V. An element p of F is called primitive if it is contained in some free generating set for F. In 1936, J. H. C. Whitehead proved that a group with generators
g
1
,
…
,
g
n
{g_1}, \ldots ,{g_n}
and one relation
r
=
1
r = 1
is free if and only if the relator r is primitive in the free group on
g
1
,
…
,
g
n
{g_1}, \ldots ,{g_n}
. In tnis paper, tne question of whether there is an analogous theorem for other varieties is considered. A necessary and sufficient condition that a finitely generated, one relation algebra be free is proved for any Schreier variety of nonassociative linear algebras and for any variety defined by balanced identities. An identity
u
(
x
1
,
…
,
x
n
)
=
v
(
x
1
,
…
,
x
n
)
u({x_1}, \ldots ,{x_n}) = v({x_1}, \ldots ,{x_n})
is called balanced if each of u and v has the same length and number of occurrences of each
x
i
{x_i}
. General sufficiency conditions that a finitely generated, one relation algebra be free are given, and all of the known results analogous to the Whitehead theorem are shown to be equivalent to a general necessary condition. Also an algebraic proof of Whitehead’s theorem is outlined to suggest the line of argument for other varieties.