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Given $\Big\{A_i \in \mathbb{R}^{m \times n}, b_i \in \mathbb{R}^{m \times 1}\Big\}_{i=1}^d$, we wish to compute $X \in \mathbb{R}^{n \times 1}$ such that, $$A_i X = b_i \;\; \forall i$$

Are there known consistency results for such joint systems of equations? Any pointers to relevant materials will be highly appreciated.

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