Can you solve these 2 equations with 3 variables and prove the solutions?

[blue_leaf77]

They are not axis, they are vectors whose number turns out to be three. In that video the solution turns out to be a sum between a fixed vector and a linear combination of the vectors denoted by $\vec a$ and $\vec b$ (just for your information the last two vectors form the basis in the so-called null space of the coefficient matrix). In your problem the solution is a sum between a fixed vector $(-150,250,0)^T$ and a linear combination of a single vector $(5/3,-8/3,1)^T$ (the last vector is a basis for the null space of the coefficient matrix in your problem).

 

[Buzz Bloom]

Hi @Nicola276:

I agree with Delta.

Something tells me that the OP wanted to say that x,y,z are positive integers.

Equations seeking integer solutions are called Diophantine equations.

There is a discussion of these equations in

https://en.wikipedia.org/wiki/Diophantine_equation .​

You may also find the following helpful. It is about a method for solving a single linear Diophantine equation.

http://www.wikihow.com/Solve-a-Linear-Diophantine-Equation​

Your problem involves two equations, but it is easy to transform it into one by eliminating one variable as shown by SammyS in post #28.

Regards,
Buzz