The main purpose of this work is to provide the general solutions of a class of linear functional equations. Let nge 2 be an arbitrarily fixed integer, let further X and Y be linear spaces over the field {mathbb {K}} and let alpha _{i}, beta _{i}in {mathbb {K}}, i=1, ldots , n be arbitrarily fixed constants. We will describe all those functions f, f_{i, j}:Xtimes Yrightarrow {mathbb {K}}, i, j=1, ldots , n that fulfill the functional equation f∑i=1nαixi,∑i=1nβiyi=∑i,j=1nfi,j(xi,yj)xi∈X,yi∈Y,i=1,…,n.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned}&f\\left( \\sum _{i=1}^{n}\\alpha _{i}x_{i}, \\sum _{i=1}^{n}\\beta _{i}y_{i}\\right) = \\sum _{i, j=1}^{n}f_{i, j}(x_{i}, y_{j}) \\\\&\\quad \\left( x_{i}\\in X, y_{i}\\in Y, i=1, \\ldots , n\\right) . \\end{aligned}$$\\end{document}Additionally, necessary and sufficient conditions will also be given that guarantee the solutions to be non-trivial.