One of the questions in my text book asks this:

The set of all pairs of real numbers of the form $\displaystyle (1,x) $ with the operations $\displaystyle (1, y) + (1, y') = (1, y + y') $ and $\displaystyle k(1,y) = (1, ky)$. The book says this is vector space, but when I check this axiom $\displaystyle (k+m)u = ku + mu$

Here is my work:

$\displaystyle

(c + k)(1,x) = (1, (c+k)x) = (1, cx + kx)

$

$\displaystyle

c(1,x) + k(1,x) = (1,cx) + (1,kx) = (2, cx+kx)

$

Where am I going wrong?

Thanks