Denote by PS(α) the image of the Piatetski-Shapiro sequence n → ⌊n α ⌋, where α > 1 is non-integral and ⌊x⌋ is the integer part of x ∈ R. We partially answer the question of which bivariate linear equations have infinitely many solutions in PS(α): if a, b ∈ R are such that the equation y = ax + b has infinitely many solutions in the positive integers, then for Lebesgue-a.e. α > 1, it has infinitely many or at most finitely many solutions in PS(α) according as α < 2 (and 0 ≤ b < a) or α > 2 (and (a, b) = (1, 0)). We collect a number of interesting open questions related to further results along these lines.