Why is $-a+b$ always equal to $b-a$?

Question

Can someone elaborate on what rules underlie the rule

$-a+b=b-a\;?$

Is it associative properties of multiplication and addition? For example,

$-a+b=b-a$

$(-1)(a)+b=b-a$

$b+(-1)(a)=b-a$

$b-a=b-a$

Are there any exceptions to $-a+b=b-a$?

Answer 1

We don't have to invoke any properties of multiplication to explain why $-a+b=b-a.$ We don't have to invoke associativity (which refers to different ways of grouping three operands) either. We merely have to know that addition is commutative (i.e., $x+y=y+x$) and every element $z$ has an additive inverse ($-z$) such that $z+ (-z) = 0.\;$ We define subtraction by $x-z=x + (-z)$. In any system where those assumptions hold, it is always true that $(-a) + b = b + (-a) = b -a$.

Answer 2

The commutative property of addition.

$$x+y = y+x$$.

Set $x = -a$ and $y=b$.

There are no exceptions.