$$6x-y+4z=12$$
$x=\frac{y-4z+12}{6}$
$$6x+4z=12+y$$
$$6x=12+y-4z$$
$$6x=y-4z+12$$
$$\frac{6x}{6}=\frac{y-4z+12}{6}$$
$$x=\frac{y-4z+12}{6}$$
$$x=\frac{y}{6}-\frac{2z}{3}+2$$
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$y=6x+4z-12$
$$-y+4z=12-6x$$
$$-y=12-6x-4z$$
$$-y=12-4z-6x$$
$$\frac{-y}{-1}=\frac{12-4z-6x}{-1}$$
$$y=\frac{12-4z-6x}{-1}$$
$$y=6x+4z-12$$
