Question

the easiest way to solve a set of linear eqns in 3 variables using matrix or determinents

Answer 1

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.

Answer 2

Cramer's Rule

Answer 3

best way is to find through Cramer's Rule

Answer 4

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.