Cauchy functional equation three variables

If I have function from $R^3$ to $R$ satisfying

$f(x_1,x_2,x_3)+f(y_1,y_2,y_3) = f(x_1+y_1,x_1+y_2,x_3+y_3)$

is it necessarily linear?

$f(z_1,z_2,z_3) = \lambda _1 z_1+\lambda _2 z_2+\lambda _3 z_3$

Wasn't sure if this was a direct consequence of Cauchy's theorem or not.