Let $F$ be a field. Show that if $x, y, z \in F $ and $x+z = y+z$ then $x = y.$

Question

Here's my attempt:

$(y + z) + (-(y + z)) = 0$

$(y + z) + (-(x + z)) = 0$

$y + z + (-x) + (-z) = 0$

$y + (-x) = 0$

Therefore -x is the negative of y.

Using the last statement how can I prove that x = y?

Answer 1

In fact you don't need the full field structure here; you just need $+$ to be associative, to have an identity, and for $z$ to have an inverse. (That is, $+$ is a group operation.)

Indeed, simply add $-z$ (which is the notation I'll use for the additive inverse of $z$) on the right of both sides: $(x+z)+(-z) = (y+z)+(-z)$. By associativity, we can shuffle the brackets so that we have $x+(z+(-z)) = y+(z+(-z))$; that is precisely $x+e = y+e$, which is to say that $x=y$.