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Superposition Theorem

The current or voltage in any linear circuit containing multiple independent sources is the algebraic sum of the voltages or currents produced by each source acting independently.
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The statement of the theorem

Let N\mathcal{N} be a linear electrical network characterized by its admittance matrix Y∈RN×N\mathbf{Y} \in \mathbb{R}^{N \times N}, where NN is the number of independent nodes. Let s={s1,s2,…,sK}\mathbf{s} = \{\mathbf{s}_1, \mathbf{s}_2, \dots, \mathbf{s}_K\} be a set of KK independent sources, where sk\mathbf{s}_k is the source vector associated with the kk-th source. The total source vector is S=∑k=1Ksk\mathbf{S} = \sum_{k=1}^{K} \mathbf{s}_k. The voltage vector v\mathbf{v} at the nodes due to the combined sources S\mathbf{S} is given by the solution to the linear system Yv=S\mathbf{Y} \mathbf{v} = \mathbf{S}. By the Superposition Theorem, the response v\mathbf{v} is the sum of the responses vk\mathbf{v}_k generated by each source sk\mathbf{s}_k acting independently: v=∑k=1Kvk\mathbf{v} = \sum_{k=1}^{K} \mathbf{v}_k. Formally, this implies that the solution vector v\mathbf{v} satisfies: v=Y−1(∑k=1Ksk)=∑k=1K(Yvk)Y−1sk\mathbf{v} = \mathbf{Y}^{-1} \left( \sum_{k=1}^{K} \mathbf{s}_k \right) = \sum_{k=1}^{K} \left( \mathbf{Y} \mathbf{v}_k \right) \mathbf{Y}^{-1} \mathbf{s}_k
Source: Wikipedia