Why is that when we are adding to both sides of an equation we don't add to every single term?

We have all heard the age old expression of what you do to one equation, you must do this to the other equation. Lets take an example equation of $$3x - 2 = 4x +5$$ when solving this equation you would add two to both sides and get $$3x = 4x + 7$$ my question is why is that when we are adding to both sides that we don't add to every single term, but when we do a multiplication or division operation we would multiply / divide every term. Also another question, in essence when we are doing something to both sides can we treat each side of the equation like a term. So in essence for the equation when we are dividing by both sides is it basically: $$1/2(3x-2) = 1/2(4x+5)$$, meaning when we do something to both sides do we treat the equation like a term? What exactly is happening behind the scenes? Why is it that you multiply every term but you don't add every single term?

• You always treat the original quantities as one term. Compare for example $\,(3x-2)\color{red}{+2}=(4x+5)\color{red}{+2}\;$ to $\;(3x-2)\color{red}{\,\times\,2}=(4x+5)\color{red}{\,\times\,2}$. It's just that, when you add, the parentheses are (usually) unnecessary. – dxiv Sep 26 '18 at 22:58
• would another person like to confirm this, if this is true it would make solving complex equations a lot easier – Marlon Henderson Sep 26 '18 at 23:27
• It is certainly true. – YiFan Sep 27 '18 at 7:22
• @MarlonHenderson I recommend you avoid using the word "complex" because it is usually reserved for a special type of numbers of "complex numbers" (Wikipedia for more). – Felix Fourcolor Sep 27 '18 at 8:36

2 Answers

I am answering this question: “Why is that when we are adding to both sides that we don't add to every single term, but when we do a multiplication or division operation we would multiply / divide every term [...] Why is it that you multiply every term but you don't add every single term?”

The short answer is that multiplication and division are special because of the distributive law: they distribute over addition and subtraction.

In general, when you apply an arithmetic operation to both sides of an equation, you do not apply it to each “term” on each side (assuming the expressions have terms at all — parts that are added or subtracted).

For example, if $$a+b=c$$ and you want to “square both sides,” you do not square each individual term to get $$a^2+b^2=c^2$$. Rather, you square the entire expression: $$(a+b)^2=c^2$$. The reason these two equations are not equivalent is that exponentiation doesn’t distribute over addition and subtraction. Most arithmetic operations are like this. Taking the absolute value, taking the sine, taking the square root — none of these operations distributes over addition and subtraction. Therefore, when you apply one of these operations to both sides of an equation, you cannot apply them to each individual term.

But the operations of multiplication and division are special. They do distribute over addition and subtraction. Multiplying both sides of $$a+b=c$$ by $$5$$ gives $$5(a+b)=5c$$. But the left side is in fact $$5a+5b$$ because of the distributive property.

In summary: you may only apply an operation to each individual term of an expression if the operation distributes over addition and subtraction.

Caveat lector: most operations do not have such a distributive property. That includes the operations of addition and subtraction themselves. Adding $$1$$ to $$a+b$$ is not the same as $$a+1+b+1$$, and that’s because addition does not distribute over itself.

symplectomorphic provided an excellent answer. However, I want to add some visuals and word it a little differently hoping that it maybe easier to understand.

The key point to understand an equation is to not think about it as a formula, but a balanced scale.

Imagine the picture above is a top-view angle of a balanced scale. We say $$C=D$$.

Now what happens if you add $$M$$ to both sides?

You can imagined the scale must be still balanced, right? After all, both sides are added the same weight! So, $$C+M = D+M$$.

What if you divide the left side of the scale into two parts A and B. Intuitively, $$A+B = D$$.

What happens if you add $$M$$ to every term?

I think it makes lots of sense that the scale cannot be balanced: You added 2 $$M$$s on the left and only 1 the right. So $$A+M + B+M \neq D+M$$

This even makes more sense if you have lots of terms on one side and very few on another, like this:

I hope you get the picture; there's no way two sides could be still balanced if you add the same thing to every term.

But why does multiplying and dividing every term work? In short, it is a coincident. As demonstrated above, things only work well if you apply stuff to both sides, so technically, when you have $$a=b+c$$ you would turn it into $$na=n(b+c)$$, not $$na=nb+nc$$. Why writing it as $$na = nb + nc$$ is still correct has been discussed by symplectomorphic, so I won't touch that topic.

In conclusion, when dealing with equation, you want to imagine it as a scale and always stress both sides equally to keep it balanced. Some special method of stressing (like multiplying) may allow you to stress every term, but this property is special and do not occur commonly in math.