How can I prove that $a=4$ in the algebraic equation $a+3=7$ using vectors and vector properties?

Question

Consider the simple algebraic equation $a + 3 = 7$. The obvious answer is $a=4$. I was tasked with proving that $a = 4$ by using vector properties. It has something to do with the additive identity property and the additive inverse property. Any help would be greatly appreciated. Also, this is only regarding vectors in two dimensions. Sorry for the ambiguity, I'm pretty confused myself.

Answer 1

What you were probably asked to do is show that $a=4$ using only the axioms of a vector space (see Wikipedia). This means that you cannot substract $3$ on both sides of the equation, beacause substraction is not described in any of the axioms.

What you can do is this:

Every element of a vector space has an additive inverse, so you know that $-3$ exists. This means that we can write: $a+3+(-3)=7+(-3)$. The associativity of addition allows us to say that this is equivalent to $a+0=4$. By definition of $0$ (and possibly by commutativity of addition) this is equivalent to $a=4$.

Now you have solved the equation using only the basic axioms of a vector space.