How to solve nonlinear multivariable systems of equations?

Question

Rather than try to explain it with words I am just going to post an example of what I am trying to do.

Solve: $$\begin{align*} a2^x+b3^y&=42\\ a4^x+b6^y&=156\\ a6^x+b9^y&=342\\ a8^x+b12^y&=600 \end{align*}$$ for $a, b, x, y$ where $a, b, x, y$ can be any real number.

The answer to this example is $x=1, y=2, a=3$, and $b=4.$ (I know this because I made this example up myself).

There are $4$ variables and $4$ equations, so via degrees of freedom there should only be 1 solution set. But without knowing any of $a, b, x,$ or $y$ in advance or guessing HOW do you solve this problem?

To give a little background I am trying to develop an estimating system using parametrics and I can't assume its a linear system.

Answer 1

Assuming the equations to be $$f_i=a\,\alpha_i^x+b\,\beta_i^y-\gamma_i=0 \tag i$$ I should first eliminate $a$ and $b$ using $(1)$ and $(2)$. This then reduces the problem to two variables and we cannot eliminate more.

Now, I should try to minimize $$\Phi(x,y)=f_3^2+f_4^2$$ It is more than probable that, if the $\alpha_i$'s and $\beta_i$'s are in arithmetic progressions, we could arrive to further elimination of $y$ as a function of $x$. In the specific problem you posted, we can get from $(4)$ $$y=\frac{\log \left(\frac{2 \left(13\ 2^x-50\right)}{7\ 2^x-26}\right)}{\log (2)}$$ and then $(3)$ is the only equation to be solved for $x$.

Answer 2

a and b can be eliminated in following way:

Step 1: Choose any two equations and solve them for a and b. So, you will have a=f1(x,y) and b=f2(x,y)

Step 2: Consider remaining two equations and again solve them for a and b. So, you will again have a=f3(x,y) and b=f4(x,y) in similar manner as step-1 above.

Step 3 : From above step-1 and step-2, we equate f1(x,y)=f3(x,y) and f2(x,y)=f4(x,y)

Step 4 : Now we have 2 equations and two variables x and y which can be solved easily further.