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 $s=7$ and 8 TeV corresponding to integrated luminosities of 1.17 to 5.1 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ at $s=7$ TeV, and 12.2 to 20.3 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ at $s=8$ 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 $s=7$ and 8 TeV respectively. The combined $tW$ cross-sections are $16.3\pm 4.1$ pb and $23.1\pm 3.6$ pb at $s=7$ and 8 TeV respectively. For the $s$-channel cross-section, the combination yields $4.9\pm 1.4$ pb at $s=8$ TeV. The square of the magnitude of the CKM matrix element $Vtb$ multiplied by a form factor $fLV$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 $|Vtd|,|Vts|\ll |Vtb|$. All the $|fLVVtb|2$ determinations, extracted from individual ratios at $s=7$ and 8 TeV, are combined, resulting in $|fLVVtb|=1.02\pm 0.04$ (meas.) $\pm 0.02$ (theo.). All combined measurements are consistent with their corresponding Standard Model predictions.