© 2019, The Author(s). 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−1 at s = 7 TeV and 12.2 to 20.3 fb−1 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 ± 5.7 pb and 87.7 ± 5.8 pb at s = 7 and 8 TeV respectively. The combined tW cross-sections are 16.3 ± 4.1 pb and 23.1 ± 3.6 pb at s = 7 and 8 TeV respectively. For the s-channel cross-section, the combination yields 4.9 ± 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| ≪ |Vtb|. All the |fLVVtb|2 determinations, extracted from individual ratios at s = 7 and 8 TeV, are combined, resulting in |fLVVtb| = 1.02 ± 0.04 (meas.) ± 0.02 (theo.). All combined measurements are consistent with their corresponding Standard Model predictions.[Figure not available: see fulltext.].