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Can I explain why the linear equation $0x_1+0x_2+0x_3+0x_4=8$ has no solution in the following way (This is a question in my homework for elementary linear algebra, $x_i$ are variables.):

For any values of $x_1, x_2, x_3, x_4$ the left-hand side of the equation is $0$ and the right-hand side is $8$. Therefore, the linear equation has no solution.

Please comment on my solution. Are there any other extra words that I can add into my solution?

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I think what you've written is perfectly acceptable. You may want to point out that $0 \neq 8$ but I don't think it is necessary.

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  • $\begingroup$ I agree. Except in some crazy (I mean crazy) number systems, $0 \neq 8$. $\endgroup$ – user44197 Dec 30 '13 at 5:50
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    $\begingroup$ They aren't any crazier than $\mathbb{R}$, just different. $\endgroup$ – Michael Albanese Dec 30 '13 at 5:52
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8=0 in any field with characteristic 2.

Examples:

$$\mathbb{Z}_{2}$$

$$\mathbb{F}_{2}/(<x^3+x+1>)$$

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  • $\begingroup$ The former providing a mathematical version of "A stopped clock is right twice a day." $\endgroup$ – Semiclassical Jun 7 '16 at 19:38

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