For two normed spaces \(X,Y\), \(f: X\to Y\) and given scalars \(a,b,A,B\), by the general linear functional equation the author means \[ f(ax+by)=Af(x)+Bf(y),\qquad x,y\in X.\tag{1} \] Its approximate solutions are defined by \[ \|f(ax+by)-Af(x)-Bf(y)\|\leq\varphi(x,y),\qquad x,y\in X\tag{2} \] with some control mapping \(\varphi\). For some particular forms of \(\varphi\), the hyperstability is proved, e.g., each solution of (2) is actually a solution of (1).