That works as far as it goes. You have shown existence.
Now show uniqueness:
I am to prove the following:
If a, b, c are real numbers s.t. a & c are NOT 0, then there is a unique number x s.t. x/a + b/c = 1.
Ok, this is what I have:
Let a, b, c be reals s.t. x/a + b/c = 1 and a & c are NOT 0.
(x * 1/a) + (b * 1/c) = 1
a[ (x*1/a) = 1 - (b * 1/c) ]
x = a [ 1 - (b *1/c)]
Since b*1/c is an element of R & 1-b/c is also and element of R,
then a(1-b/c) is also an element of R.
So x is an element of R.
This seems pretty straight forward, but I just want to make sure that I am not approaching this wrong or am missing something in my proof. Thanks.