= Solution
For each nonzero <integer> tuple $k=(k_1,\ldots,k_{n+1})$, put
$$
H_k=\left\{x\in\mathbb R^n:\sum_{j=1}^nk_jx_j=k_{n+1}\right\}.
$$
If its first $n$ coordinates are all zero, then $k_{n+1}\ne0$ and $H_k$ is empty. Otherwise it is a <closed> proper <affine hyperplane>. It has empty interior: from any of its points, an arbitrarily small displacement in the nonzero normal direction $(k_1,\ldots,k_n)$ leaves it. Thus every $H_k$ is <nowhere dense>.
There are countably many <integer> tuples, and $\mathbb R^n$ is complete. The <Baire category theorem> therefore makes the complement of their union <dense> and nonempty. Choosing any point there gives
$$
\boxed{\sum_{j=1}^nk_jx_j\ne k_{n+1}\quad\text{for every nonzero integer tuple }k.}
$$
Equivalently, $1,x_1,\ldots,x_n$ are <linearly independent> over $\mathbb Q$: multiplying a rational relation by a common denominator produces just such an <integer> relation. This is an instance of <Baire avoidance of countably many affine hyperplanes>.
Back to article page
