# How can I systematically solve a system of non-linear equations using a standard graphing calculator?

In analytical chemistry, when we have to consider more than one equilibrium, the problem can be reduced to solving a system of non-linear simultaneous equations by applying the mass and charge balance equations.

Conventionally, the equations are converted to a polynomial equation through simple substitution and then subsequently solved using an online equation solver. So, I was wondering if there's an easier way to solve the system of equations directly (i.e., without manipulating to a polynomial) using a graphing calculator.

I know this just sounds like I'm being lazy, but it would definitely save a lot of time considering that I'm used to dealing with 6 or more equations at a time. For reference, the graphing calculator I'm referring to is the TI-nspire CX II.

Any insight into this would be greatly appreciated. Thanks for reading!

• You could try making a program in your calculator using Newton's method (Jacobian matrix iteration) with an initial guess. I made one for a TI-84 calculator some years ago (older version of your calculator), so it should be a little easier in your case. Commented Feb 3, 2023 at 17:02
• @Sam202 The Newton method is very efficient when it converges, but it's convergence may be even for single independent variable tricky at unlucky choice of starting point and therefore of combinations of respective derivatives. (convergebce condition |f.f''| < (f')^2 ) Commented Feb 4, 2023 at 8:56
• @Poutnik That is correct. Although not completely practical and far from perfect, one could manipulate convergence condition within the iteration loop to obtain an approximate answer, or perhaps re-run the program with a different initial guess if no convergence is obtained. I am guessing OP is reluctant to use Excel or alike because it is not allowed in the exam, whereas his calculator is. Commented Feb 4, 2023 at 17:38

Have you considered conversion of the set of nonlinear equations to an optimization problem?

If we have set of equations: \begin{align} L_1 &= R_1\\ L_2 &= R_2\\ ..&.. \\ L_n &= R_n \end{align}

then solve numerical optimization job = minimization of the value of the custom function:

$$U = \sum_{i=1}^{n}{\left(L_i - R_i \right)^2} = \mathrm{min}$$

$$L_i$$ and $$R_i$$ are the respective left and right sides of the equation No $$i$$.

I assume a set of 3 and more equations cannot be solved easily by a graphic way. OTOH many such calculators can be tasked by some simple automation program.

With handiness advantage, one can use the MS Excel optional add-in Solver, designed exactly for similar cases. It finds values of input parameters for minimization of the dependent target value. In case of problem with relative scaling of variables across many orders, there is option of scaling adaptation.