  This paper presents the combinations of single-top-quark production
cross-section measurements by the ATLAS and CMS Collaborations, using data from
LHC proton-proton collisions at $\sqrt{s}=7$ and 8 TeV corresponding to
integrated luminosities of 1.17 to 5.1 fb$^{-1}$ at $\sqrt{s}=7$ TeV, and 12.2
to 20.3 fb$^{-1}$ at $\sqrt{s}=8$ TeV. These combinations are performed per
centre-of-mass energy and for each production mode: $t$-channel, $tW$, and
$s$-channel. The combined $t$-channel cross-sections are $67.5 \pm 5.7$ pb and
$87.7 \pm 5.8$ pb at $\sqrt{s}=7$ and 8 TeV respectively. The combined $tW$
cross-sections are $16.3 \pm 4.1$ pb and $23.1 \pm 3.6$ pb at $\sqrt{s}=7$ and
8 TeV respectively. For the $s$-channel cross-section, the combination yields
$4.9 \pm 1.4$ pb at $\sqrt{s}=8$ TeV. The square of the magnitude of the CKM
matrix element $V_{tb}$ multiplied by a form factor $f_{\rm LV}$is determined
for each production mode and centre-of-mass energy, using the ratio of the
measured cross-section to its theoretical prediction. It is assumed that the
top-quark-related CKM matrix elements obey the relation $|V_{td}|,|V_{ts}| \ll
|V_{tb}|$. All the $|f_{\rm LV}V_{tb}|^2$ determinations, extracted from
individual ratios at $\sqrt{s}=7$ and 8 TeV, are combined, resulting in
$|f_{\rm LV}V_{tb}|=1.02\pm0.04$ (meas.) $\pm0.02$ (theo.). All combined
measurements are consistent with their corresponding Standard Model
predictions.
