Solution to 4 equations and 3 unknowns?

I have these 4 equations:

and I am looking for the values of Px, Py and Pz, so these are equations to find the X, Y and Z coordinates (kind of see what a GPS is doing. I wanted to immitate a GPS and see how they find our coordinates.

I've been trying to solve this for a very long time with matrixes but I seem to be bugging.

I deconstructed all the ("X-P")^2 and then as I mentionned, I was attempting to do matrixes to solve all of these because all the Px, Py and Pz are supposed to be the same.

• Relevance to space exploration? – Organic Marble Feb 24 at 20:20
• @OrganicMarble equations to find the X, Y and Z coordinates (kind of see what a GPS is doing) – alana Feb 24 at 20:21
• Consider editing your question to make it clear it's space-exploration-related and not just a math problem. – Organic Marble Feb 24 at 20:22
• @OrganicMarble oh okay I'm so sorry! Thank you:) – alana Feb 24 at 20:22
• @OrganicMarble I'll ask the math community too to see if someone can help :) – alana Feb 24 at 20:25

In this case, all your equations happen to be polynomials (once you move the $$v$$ terms to the other side and square), so you can do fancy tricks with modern computer algebra tools like Gröbner bases, but this is very uncommon in the navigation and geolocation literature.