The problem is:
The points (2,-1,-2), (1,3,12), and (4,2,3) lie on a unique plane. Where does this plane cross the z-axis?
I can easily solve this problem by calculus and cross product: the equation of a plane is
Hence, this plane crosses the z-axis at z=5. However, the problem requires to be sold by the system of linear equations transformed into matrix. Even vectors have not been introduces yet. Just a matrix in echelon form and a back substitution. Hence, when I am writing three equations:
I end up with 4 variables to find. What am I missing? Is there one more linear equation possible to add?