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 +- 5.7 pb and 87.7 +- 5.8 pb at sqrt(s) = 7 and 8 TeV respectively. The combined tW cross-sections are 16.3 +- 4.1 pb and 23.1 +- 3.6 pb at sqrt(s) = 7 and 8 TeV respectively. For the s-channel cross-section, the combination yields 4.9 +- 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_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| << |V_tb|. All the |f_LV V_tb|^2 determinations, extracted from individual ratios at sqrt(s) = 7 and 8 TeV, are combined, resulting in |f_LV V_tb| = 1.02 +- 0.04 (meas.) +- 0.02 (theo.). All combined measurements are consistent with their corresponding Standard Model predictions.