I have a presentation $\langle X\mid R\rangle$ of some group $G$, and some words $W_1, ..., W_n\in F(X)$ which I know generate $G$.
How can I get a presentation for $G$ corresponding to the generators $W_1, ..., W_n$?
My idea was to form the presentation $\langle X, a_1, ..., a_n\mid R, A_i=W_i\rangle$ using Tietze transformations and then eliminate the generators of $X$. For example, consider $G=\langle x, y\mid x^{-1}y^2x=y^3\rangle$. I know that $x$ and $y^2$ generate $G$, so: $$ \begin{align*} \langle x, y\mid x^{-1}y^2x=y^3\rangle &\cong \langle x, y, t\mid x^{-1}y^2x=y^3, t=y^2\rangle\\ &\cong \langle x, y, t\mid x^{-1}txt^{-1}=y, t=y^2\rangle\\ &\cong \langle x, t\mid t=(x^{-1}txt^{-1})^2\rangle \end{align*} $$
Is this correct?