Common decision of any diophantine equation
Input data
The diophantine equation created on a matrix
Solution of system

We will continue to develop the theme of Diophantine equations and consider the following problem: Let us know some general system for solving a linear homogeneous Diophantine equation with several variables.

In our case, it looks like this

Source equation

Initial data

Common decision

2p-7q + 3 = x_3

11p-2q-1 = x_2

-5p + q + 1 = x_1

It should be a general decision to find the coefficients of the original equation.

The first thing that comes to mind is to substitute the general into the original expression, group, shorten the like and probably get the correct result.

Why "probably"?

Because such a thought did not occur to me. Want to check, it's not mine.

Another idea may consist in calculating several values ​​from a general system and solving  a system of linear equations online . Such an idea is not without meaning, but again this is not my method.

As always, I went the other way and using the created calculator Fundamental solution of system of the equations  solved this and similar problems.

 

By entering the coefficients of the general solution 6 -7 3 11 -2 -1 -5 1 1 we get a wonderful and most importantly correct answer

-x_ {3} + 10 * x_ {2} + 13 * x_ {1} = 39

One more example

General system of the equations 

Исходные данные
The diophantine equation created on a matrix 
Диофантово уравнение

 

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