This　paper　is　concerned　with　a　stabilized　finite　element　method　based　on　two　local　Gauss　integrations　for　the　two-dimensional　non-stationary　conduction-convection　equations　by　using　the　lowest　equal-order　pairs　of　finite　elements.　This　method　only　offsets　the　discrete　pressure　space　by　the　residual　of　the　simple　and　symmetry　term　at　element　level　in　order　to　circumvent　the　inf-sup　condition.　The　stability　of　the　discrete　scheme　is　derived　under　some　regularity　assumptions.　Optimal　error　estimates　are　obtained　by　applying　the　standard　Galerkin　techniques.　Finally,　the　numerical　illustrations　agree　completely　with　the　theoretical　expectations.