I asked my Calculus Teacher this and he had no answer, we were talking about linear equations just to brush up on grade 12 maths and I never thought to ask but is it at all possible to describe an equation like x=3 in terms of y? There is no y term in the equation but is there any crazy math to describe it?
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J. W. Tanner
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Welcome to Mathematics Stack Exchange. For a vertical line, $y$ could be anything. I suppose you could say $x=3+0y$ – J. W. Tanner Feb 17 '20 at 03:57
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@J.W.Tanner that does technically answer my question but I was looking for something with y = some expression. – Carter Tomlenovich Feb 17 '20 at 04:03
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There's no function $f(x)$ such that $y=f(x)$ describes a vertical line. – Ben W Feb 17 '20 at 04:06
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$0y=x-3{}{}{}$. – Feb 17 '20 at 04:11
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How about this? $$y=0^{-(x-3)^2}+u,\quad u\in\mathbb R$$ – mr_e_man Feb 17 '20 at 04:11
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@Rahul: Nice! ${}{}{}$ – mjw Feb 17 '20 at 04:13
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@mr_e_man, what is that? – mjw Feb 17 '20 at 04:13
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@mjw - If $x\neq3$, then we have $0$ raised to a negative power, which is undefined (or infinite). If $x=3$, then we have $0^0=1$ so $y=1+u$ which is any real number. – mr_e_man Feb 17 '20 at 04:15
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@mr_e_man yes! Thank you. I love this solution! – Carter Tomlenovich Feb 17 '20 at 04:23
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Every equation for a line in the $xy$-plane can be put in "standard form" $Ax+By=C$. Vertical lines in particular always can be written in the form $1x+0y=C$. In this case it's usually nicer just to write $x=C$, but you can always interpret it as having an "invisible" $0y$ term.
Ben W
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But not every "standard form" describes a line. What about $A=B=0, $and $C\ne0$? – mjw Feb 17 '20 at 04:11
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@Ben_W, Was not contradicting your answer. Just observed that there are choices for ${A,B,C}$ that do not describe a line. – mjw Feb 17 '20 at 18:13