# How could I solve a system of equations of the form $xn_i^y+z=m_i$

How could I solve the following system of equations:

$$xn_1^y+z=m_1$$ $$xn_2^y+z=m_2$$ $$xn_3^y+z=m_3$$ where $n_k$ and $m_k$ are parameters?

I am looking for an exact answer, but if it is not feasible, then a rough description of a numerical approach would be appreciated. Thank you.

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You will have to change your tags and question because this is not linear algebra problem. Linear algebra deals with systems of linear equations; linear equation have a form $a_1x_1 + ... a_nx_n=b$ where $a_i$ and $b$ are given and you search for $x_i$. Problem you described in nonlinear. – Trismegistos Feb 20 '12 at 19:36
@Trismegistos: I agree; yet I can't find an existing tag that fits... – Arturo Magidin Feb 20 '12 at 19:40
@ArturoMagidin I think real-analysis would be better tag and it should certainly have numerical-methods tag. – Trismegistos Feb 20 '12 at 19:43
@Trismegistos: Sounds good; I assume you cannot do a tag edit due to your reputation level, so I've gone ahead and done it. – Arturo Magidin Feb 20 '12 at 19:45

After eliminating $x$ and $z$, you are left with $(m_2 - m_3) n_1^y + (m_3 - m_1) n_2^y + (m_1 - m_2) n_3^y = 0$. Unfortunately this won't have closed-form solutions in general; you'll have to rely on numerical methods. The standard numerical equation-solvers could be used. You might try Newton's Method if you're doing it yourself, but it's usually better to use existing tools if you have them available, e.g. fsolve in Maple.