Obtaining the equations after scaling and shifting?

I am thinking about this coordinate shift. Suppose we have the two equations as below, where $$x_0,y_1, x,y,\bar x,\bar y$$ are variables while others are constants:

\begin{aligned} \bar{x_0} &= x^+ + a\lambda_{k}x_{0} + b(y_{1} - y^-) + \ldots \\ \gamma^{-k}\bar{y_1} &= \mu + c\lambda^k x_{0} + f_{11}\lambda^k x_{0}(y_{1} - y^-) + d(y_{1} - y^-)^2 + \ldots \end{aligned}

Now, we employ a coordinate shift: $$x_0 - x^+ \mapsto x~~, \quad y_1 - y^- \mapsto y$$

and then an additional small shift

$$x \mapsto x + \frac{bf_{11}}{2d} \lambda^k ~~, \quad y + \frac{f_{11}x^+}{2d}\lambda^k \mapsto y$$

I am thinking how this kills the constant term in the first equation and the term linear in $$y$$ in the second equation so that the first two equations now become

\begin{aligned} \bar{x} &= by + \ldots \\ \gamma^{-k}\bar{y} &= \mu - \gamma^{-k}y^- + c\lambda^k x^+ + c\lambda^k x_{0} + f_{11}\lambda^k xy + dy^2 + \ldots \end{aligned}

Is it intuitive to start from the first two equations and apply the coordinate change and shift to get the last two equations? I am trying but not obtaining them, like I got the terms of the first equation but I keep on having the $$x_{0}$$ term?