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How to solve this system of two equations with three unknowns with Cramer's formula: $$\begin{cases} x-y+3z&=&0\\ 2x-2y+6z&=&0 \end{cases}$$ I tried to find $\Delta xy$, $\Delta xz$, $\Delta yz$ and all of them are equal to $0$, so I think the system doesn't have any solution.

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Both equations are multiple of each other. Hence, all $(x,y,z)$ with $x=y-3z$ are valid solutions.

You could write down all solutions like $$v = (y-3z,y,z)^T=y(1,1,0)^T+z(-3,0,1)^T$$ where $y,z \in R$ are arbitrary constants.

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