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What is the proper notation when solving one equation by inserting another known relationship?

$$y=x^2, y+2x^2=1$$$$y+2x^2=1\Leftrightarrow x^2+2x^2=1 \Leftrightarrow x=\pm\sqrt{\frac{1}{3}}$$

I assume the first $\Leftrightarrow$ is wrong since we are inserting a relationship, and the $y+2x^2=1$ equation is not logically equal to $x^2+2x^2=1$ by it self.

Edit: I guess what I'm asking is: Is there a way of notating $f(x),g(x)\Rightarrow h(x)$ cleanly?

Erik Eriksson
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2 Answers2

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\begin{align}&&y=x^2 \;&\text{ and }\; y+2x^2=1\\&\iff &y=x^2 \;&\text{ and }\; x^2+2x^2=1 \\&\iff &y=x^2 \;&\text{ and }\; x=\pm\sqrt{\frac{1}{3}}\\&\iff &y=\frac13 \;&\text{ and }\; x=\pm\sqrt{\frac{1}{3}}.\end{align}

ryang
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  • ok. So the "and" is enough to justify the logically equivalence. Thank you. – Erik Eriksson Nov 01 '21 at 13:00
  • @ErikEriksson Yes; just to convince yourself, do verify the chain of equivalences by checking backwards as well as forwards. This long-winded presentation isn't my favourite, but I'm merely addressing your request (equation solving implicitly means that the logic is a chain of "$\iff$", although in practice, most people just work using implicit "$\implies$" and then, if necessary, check the final line for extraneous solution). – ryang Nov 01 '21 at 13:01
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Why do you need any notation?

Since $y=x^2$ and $y+2x^2=1$, we have $3x^2=1$, which is equivalent to $x=\pm\sqrt{1/3}$.

Especially Lime
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  • That is certainly a good solution. I tend to fill my solutions with numbers and symbols, and that can be hard on the eye. But when I want to notate a simple equation system solution the simple symbols could be nice. Thank you for the beautifying my solution :) – Erik Eriksson Nov 01 '21 at 15:36