Is there any algorithm to find the solution of a system of 2 linear and 1 algebraic equation?

Question

I have a system such as:

$$\begin{aligned} a_1+b_1t&=u\\ a_2+b_2t&=v\\ a_3+b_3t&=f(u,v)\; \end{aligned}$$

Where a1, b1, a2, b2, a3, b3 are known constants, t, u, v are real variables and f(u,v) is an algebraic function. Is there any known algorithm to solve that system for t?

Answer 1

Using your notations, just replacing, you then have $$a_3+b_3t=f(a_1+b_1t,a_2+b_2t)=\Phi(t)$$ which makes one (implicit) equation in $t$. The solutions for $t$ would probably require some iterative numerical schemes such as Newton or other root-finders.

You may visualize the problem as the intersection of a straight line $y=a_3+b_3t$ and a curve $y=\Phi(t)$.