Is there a simple way to find the system of linear equations given the solutions? For example, find a system with 2 equations and 3 variables that has solutions (1, 4, -1) and (2, 5, 2).
I get:
x1 - x2 = -3
3x1 - x3 = 4
The method I used to get this answer was guessing and checking, but surely there is a simpler and more efficient way to go about it?
Also, are there other equations that would fit this solution set?
Thanks!
The solution set of your system of equations is not simply $\{(1,4,-1), (2,5,2) \}$.
For example, $(0,3,-4)$ is a solution. There are infinitely many others.
And we cannot obtain a system of two linear equations in three unknowns whose solution set is $\{(1,4,-1), (2,5,2) \}$. For if the system has a solution, it has infinitely many.
As to your question about producing such a system in a systematic way, let the two equations be $a_1x+b_1y+c_1z=d_1$, $a_2x+b_2y+c_2z=d_2$. The conditions are
$a_1+4b_1-c_1=d_1$, $2a_1+5b_1+2c_1=d_1$, $a_2+4b_2-c_2=d_2$, $2a_2+5b_2+2c_2=d_2$.
We have $4$ equations in $8$ unknowns, so obviously a lot of slack. One way to cut down on it is to decide that the $d_i$ will be $0$.
Choose say $a_1=0$, and solve for $b_1$ and $c_1$; Choose $b_2=0$, and solve for $a_2$ and $c_2$.