Let
$X$ and
$Y$ be two normed spaces over fields
$\mathbb{F}$ and
$\mathbb{K}$, respectively. We prove new generalised hyperstability results for the general linear equation of the form
$g(ax+by)=Ag(x)+Bg(y)$, where
$g:X\rightarrow Y$ is a mapping and
$a,b\in \mathbb{F}$,
$A,B\in \mathbb{K}\backslash \{0\}$, using a modification of the method of Brzdęk [‘Stability of additivity and fixed point methods’, Fixed Point Theory Appl.2013 (2013), Art. ID 285, 9 pages]. The hyperstability results of Piszczek [‘Hyperstability of the general linear functional equation’, Bull. Korean Math. Soc.52 (2015), 1827–1838] can be derived from our main result.