For a real constant $α$, let $π_3^α(G)$ be the minimum of twice the number of $K_2$'s plus $α$ times the number of $K_3$'s over all edge decompositions of $G$ into copies of $K_2$ and $K_3$, where $K_r$ denotes the complete graph on $r$ vertices. Let $π_3^α(n)$ be the maximum of $π_3^α(G)$ over all graphs $G$ with $n$ vertices. The extremal function $π_3^3(n)$ was first studied by Győri and Tuza [Decompositions of graphs into complete subgraphs of given order, Studia Sci. Math. Hungar. 22 (1987), 315--320]. In a recent progress on this problem, Král', Lidický, Martins and Pehova [Decomposing graphs into edges and triangles, Combin. Prob. Comput. 28 (2019) 465--472] proved via flag algebras that $π_3^3(n)\le (1/2+o(1))n^2$. We extend their result by determining the exact value of $π_3^α(n)$ and the set of extremal graphs for all $α$ and sufficiently large $n$. In particular, we show for $α=3$ that $K_n$ and the complete bipartite graph $K_{\lfloor n/2\rfloor,\lceil n/2\rceil}$ are the only possible extremal examples for large $n$.