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For example with the equation $$5x-4 = 2x +5$$ Is the accepted theory that you think of this equation in terms of: $(5x-4) = (2x+5)$ when you are doing an operation to both sides. Lets say I wanted to multiply by $3$, I would do: $$3(5x-4) = 3(2x+5)$$ Is there ever a scenario that would break the rule of thinking of equations in a matter of each side being a whole term? Is this the accepted way of solving equations?

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    $\begingroup$ The name to use is expression rather than term. One expression is equal to another. Yes, if you wish to multiply both sides by $3$ you would do so to each expression. To answer your question of if you would ever not think of each side as a larger piece, sure... consider situations where you want to multiply by $1$ or add $0$. Take the expression $\frac{n-1}{n+1}$. You could rewrite this as $\frac{n-1 +0}{n+1}=\frac{n-1+2-2}{n+1}=\frac{n+1-2}{n+1}=\frac{n+1}{n+1}+\frac{-2}{n+1}=1-\frac{2}{n+1}$. If I happened to have that expression equal to something else, I only need to modify first $\endgroup$ – JMoravitz Sep 27 '18 at 4:27
  • $\begingroup$ your idea is correct, when we apply such rule we think that x is some fixed number thus 5x-4 and 2x+5 are also some fixed numbers thus what you did here you say : "Suppose 5x - 4 = 2x+5 for some real number x, then 3*(5x-4) = 3*(2x+5)" here you apply a theorem that says : "for any real numbers a,b if a = b then ac = bc for any real number c" thus having 3(5x-4)=3(2x+5) $\endgroup$ – famesyasd Sep 27 '18 at 4:27
  • $\begingroup$ This is essentially the same question you posted a few hours ago. You could have just edited that one, instead. $\endgroup$ – dxiv Sep 27 '18 at 5:10
  • $\begingroup$ Welcome to MathSE. This tutorial explains how to typeset mathematics on this site. $\endgroup$ – N. F. Taussig Sep 27 '18 at 8:31
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A deeper perspective on what you're missing is the notion of substitution.

Here is a toy example.

Suppose that

$$a+\color{blue}{b}=c\tag{1}$$

And suppose as well that

$$\color{blue}{b}=\color{blue}{d+e}\tag{2}$$

Then, because $b$ equals $d+e$, we may substitute $d+e$ for $b$ in the first equation (indeed, in any equation containing $b$), giving

$$a+\underbrace{\color{blue}{d+e}}_{b}=c\tag{3}$$

Thus, informally speaking, you might say that in moving from (1) to (3) we have "manipulated one term in the left side of equation (1) without manipulating the whole left side, and without manipulating the right side at all." This is correct, and a perfectly valid way of reasoning.


On the other hand, apparently you have heard it said there is a "rule" that you must "always do the same thing to both sides of an equation." This is both imprecise and wrong. As we saw above, it is logically valid to "do something" to one side of an equation -- even part of one side of an equation -- without doing something to the other side.

What the rule means to say is much wordier: if you apply an arithmetic operation (addition, subtraction, multiplication, division, exponentiation, substitution into a trig function, whatever) to the entire expression on one side of an equation, you must perform that operation on the entire expression on the other side, too, in order to preserve the equality.

Even more precisely: if $a=b$, then $f(a)=f(b)$, no matter the function $f$.

In this case, we are substituting $a$ and $b$ as inputs into some function and deducing that the outputs of the function are equal. For more on this perspective as it relates to equation-solving, see my answer here.


A postscript: one way you can tell the "rule" is wrong is that it says to always do something, period.

But mathematical theory doesn't dictate behavior ex cathedra: it doesn't issue statements of the form "do this." That's what pseudocode for a computer program says; it's not what mathematics says.

Rather, mathematics says do this if you want to be consistent, or draw a certain conclusion, or whatever. What you do in mathematics depends on your goal. In order to obtain such-and-such a goal, you should do such-and-such.

You will be a better mathematical thinker if you constantly scrutinize procedural or imperative commands from your teachers -- do such-and-such -- for their real mathematical meaning. Procedures are only important to the extent they help us achieve a certain goal.

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Yes, this is the most common way of thinking of the two sides of an expression. There is no case when you can have an equality, in your case $5x-4=2x+5$, and multiply only one part of each side by a constant. As for a reason why this is impossible, if one does this it will change the solution set of the expression, which is essentially another way to identify the expression.

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