2. For a point $(x, y)$ on the line $y=2 x+1$, after translating it by the vector $\boldsymbol{a}\left(x_{0}, y_{0}\right)$, we get a point $\left(x_{1}, y_{1}\right)$ on the line $y=2 x-1$, with $x+x_{0}=x_{1}, y+y_{0}=y_{1}$. Then the value of $2 x_{0} - y_{0}$ ( ).
(A) equals 1
(B) equals 2
(C) equals 3
(D) cannot be determined, depends on the specific values of $x_{0} 、 y_{0}$
2.B.

Substitute $x+x_{0}=x_{1}, y+y_{0}=y_{1}$ into the line $y=2 x-1$ to get $y+y_{0}=2\left(x+x_{0}\right)-1$, which is
$$
y=2 x+\left(2 x_{0}-y_{0}-1\right) \text {. }
$$

Equation (1) should be the line $y=2 x+1$. Comparing the two equations, we get
$$
2 x_{0}-y_{0}-1=1 \text {. }
$$

Thus, $2 x_{0}-y_{0}=2$.