I know that if I have an equation of the form $ax + by + cz = 0$, I can calculate the basis in $R^3$. For example if I have a linear equation $x - 2y + 5z = 0$, then from mu understanding the basis are ${(2,1,0)(1,-2,-1)}$.
But now, let say I have a system of non homogeneous equation
$x - 2y + 5z = 4$,
$3x + 1.1 y + 2z = 7$
How can I find the basis of this system? Do I have to find the basis of the homogeneous form of the equation? Then what is the role of the constant terms on LHS?
From my understanding, the constant terms just provide the bias or shift to the basis from the origin but still I cannot wrap my head around this!
My ultimate goal is to project a circle or an ellipse on this surface. Please help me. Thanks.