Representing that I'm simplifying both sides of an equation

Question

If I'm solving an equation and want to show that I'm simplifying both sides of an equation without writing it in words, how do I do it?

If I'm only simplifying one side of an equation, it is simple: $$\begin{align} 5&=\frac{4}{3}x+7x\\&=\frac{4}{3}x+\frac{3}{3}7x\\&=\frac{4}{3}x+\frac{21}{3}x\\&=\frac{25}{3}x\\\Leftrightarrow{x}&=\frac{3}{25}5\\&=\frac{15}{25}\\&=0.6\end{align}$$ All I have to do is make sure that the left side of the equation is the expression I'm not simplifying, and everytime I do something to both sides, I use the bi-implication symbol $\Leftrightarrow$.

But if I were to simplify both sides of an equation like here: $$\begin{align}\\\frac{1}{3}x+\frac{12}{3}x+4-4&=2-4\\\frac{13}{3}x&=-2\end{align}$$

How do I show that the two equations are related? Is the bi-conditional $\Leftrightarrow$ appropriate to use, or should I use another symbol?

Answer 1

$$\begin{align} 5&=\frac{4}{3}x+7x\\&=\frac{25}{3}x\\\Leftrightarrow{x}&=\frac{3}{25}\times5\\&=0.6\end{align}$$ All I have to do is make sure that the left side of the equation is the expression I'm not simplifying, and every time I do something to both sides I use the bi-implication symbol ⇔.

Then you're actually using ⇔ wrongly, because your explanation above indicates that you'd write $$5=\frac{25}3x\\\color\red⇔25=\left(\frac{25}3x\right)^2.$$ The above ⇔ says, incorrectly, that the two lines have the same solution set, yet $-0.6$ satisfies the second but not the first line.

$$\begin{align}\\\frac{1}{3}x+\frac{12}{3}x+4-4&=2-4\tag1\\\frac{13}{3}x&=-2\tag2\end{align}$$

How do I show that the two equations are related? Is the bi-conditional $\Leftrightarrow$ appropriate to use, or should I use another symbol?

This presentation is fine, because it is tacit that line (1) implies line (2), as intended. Do not write ⇔ just to indicate that line $(2)$ is a simplification of both sides of line $(1):$ that red ⇔ above says, incorrectly, that the two equations around it imply each other, contrary to your intention.

Short answer: typically, you mean merely that line (m) implies/⇒ line (m+1), in which case the stronger symbol ⇔ is not generally appropriate, because it additionally says that line (m+1) implies line (m). Omitting the intended connective symbol ⇒ is fine, as it is inferred by the reader.

Answer 2

The biconditional is not per se inappropriate to use, but it is, in almost all such instances, superficial. It should be clear that one line lying under another follows from the line above. This doesn't need to be notated with any logical connective. But you are correct that $\Rightarrow$ or $\iff$ (depending on the reversibility of the implication) would correctly notate the implication.

Any rule of inference (i.e., step in logic or algebraic) that connects two lines should be implied and sufficiently obvious to an appropriate reader that it could be understood without being made explicit. And that should also be true for any other such lines. In fact, all of your lines have an implied manipulation.

Good mathematical writing preempts what its intended reader knows and is able to skip minor points that can be understood as implied. If anything would not be clear to the reader, it should be spelled out explicitly.

Answer 3

When simplifying equations, I would personally use the $\implies$ symbol. The $\iff$ symbol is usually reserved for connecting different statements together, like saying "$\forall d\in\mathbb{Z}^+, d\not\mid x\iff x$ is prime". Simplifying is basically rewriting the same statement in simpler forms, so the $\iff$ symbol would not be appropriate. However, it is, after all, a personal preference, so you are free to use the $\iff$ symbol. It is clear that you are simplifying the equation.