  Consider a translation-invariant system of linear equations $V x = 0$ of
complexity one, where $V$ is an integer $r \times t$ matrix. We show that if
$A$ is a subset of the primes up to $N$ of density at least $C(\log\log
N)^{-1/25t}$, there exists a solution $x \in A^t$ to $V x = 0$ with distinct
coordinates. This extends a quantitative result of Helfgott and de Roton for
three-term arithmetic progressions, while the qualitative result is known to
hold for all systems of equations of finite complexity by the work of Green and
Tao.
