Blow up algebras

Question

Consider a ring $R$, an ideal $I$ in $R$ (with generators $g_1,\ldots,g_n$). Then for some $f\in I$ the map $R\rightarrow R[x_1,\ldots,x_n]/(fx_1-g_1,\ldots,fx_n-g_n)$ should correspond to blowing up $\text{Spec}(R)$ along $I$. If I take $R=k[a,b]$ the affine plane, $I=(a,b)$, $f=a$ then I get $k[a,b][x,y]/(a x-a,ax-y)=k[a,b,x]/(ax-a)$. Shouldn't this be a polynomial ring in two variables? Geometrically the Spec is an affine chart in the blow up.