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For example, is the following a single equation or two equations?

$$ \frac{x-1}{2} = \frac{y-2}{-4} = \frac{z+3}{1}.$$

A textbook I'm looking at refers to the above as a single equation. But I would've thought that the above involves 2 equals signs and thus involves 2 equations.

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    The border between singular and plural is a bit fuzzy, since one vector equation $(x,y,z)=(1,2,3)$ is equivalent to three scalar equations $x=1$, $y=2$, $z=3$... – Hans Lundmark Apr 21 '16 at 06:47

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These are two different equations, precisely: $$ \frac{x-1}{2} = \frac{y-2}{-4} , $$ and

$$ \frac{y-2}{-4} = \frac{z+3}{1}.$$

Clearly the third $ \frac{x-1}{2} = \frac{z+3}{1}$ become a consequence of the first two. I don't know in which dimension you are, but in $\mathbb{R}^3$ these equations describe a straight line. This is only a compact form to write it, instead of a linear sistem.

Consider them separately: in other words \begin{cases} -2x+2=y-2\\ y-2=-4z-12\\ x-1=2z+6 \end{cases}

This is a linear sistem of rank 2.

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