the easiest way to solve a set of linear eqns in 3 variables using matrix or determinents
Determinents allow you to see if the system is linearly independant (or not) without needing to actually solve the system. det() = 0? i.e. 1x + 2y + 3z = 0 (eq 1) -2x + 0y -2z = 2 (eq 2) -1x + 2y + 1z = 2 (eq 3) is a system that's not linearly independant (eq 3 = eq 2 + eq 1) Therefore, there is not a unique solution for the 3 variables x,y, & z.
best way is to find through Cramer's Rule
Best way to find it is to use Gaussian elimination. Cramer's rule is computationally inefficient, but easier on the brain if you have a calculator.