Typically, we are able to solve
system of linear or nonlinear equations which are a set of simultaneous
equations (SE). Definitely, solving of a linear SE is very easy while we have
to use Newton's method to solve nonlinear SEs. The common case for both of them
is, to generate an equation for each variable. It means, we can solve an
equation with three variables, if we have three simultaneous equations or solving
of four variables needs to find a system of four simultaneous equations and so
on.

Can we solve a nonlinear equation with
many variables? Yes. In the special conditions, the answer is

positive.

The purpose of this
article is to present some examples which show us possibility to solve a
nonlinear equation with many variables where we have a good estimation for
limited domain and range of variables. The method applied is the same method
stated in article of "The Generating New Probability Theorems"
posted on link:

The experienced
physicists and engineers have usually the true speculation of domain and range
for the variables while they need to obtain precise amounts for the variables.
Therefore, this method can be useful for them.

At the first, I start by a nonlinear
equation with three variables then four variables and finally five variables.

**Example (1): Solve Circle Equation**

**When we open a calculus book, we can see the signs and footprints of Pythagoras (582 B.C – 496 B.C) everywhere. Therefore, let me start by solving of circle equation for a limited domain and range as follows:**

Consider the circle equation with
below domain:

If

x^2 + y^2 = r^2

x, y ϵ N, x, y ≤ 100

Then the range will be

r ϵ N, r ≤ 141

Now, I apply above method and get
all results of "x, y" for "r" in given range. All number of
results to "r" has been presented on below graph

In this case, total sum of possible
answers is equal to 126.

For instance, above
graph shows us, if r = 25, 50, 75, 100 then the number of results for "x, y"
are equal to 4 and if r = 65, 85 then the number of results for "x, y"
are equal to 8. The results are as follows

Now, consider the circle
equation with below domain

If

x^2 + y^2 = r^2

0 0 0 x, y ϵ N, x, y ≤ 1

Then the range will be

4 r ϵ N, r ≤ 14 1

If I apply above method, I will generate all
results of "x, y" for "r" in given range. All number of
results to "r" has been presented on below graph:

In this case, total sum of possible
answers is equal to 2068.

For instance, above graph shows us, if
r = 325, 425, 650,725,850,925,975 then the number of results for "x,
y" are equal to 14. The results are as follows:

**Example (2): Solve Circle Equation**