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There are common formulae for solutions of system of 3 linear equations as
They are usually expressed using determinant of 3x3 matrixes, which is defined as
The solutions then will be
thus each variable is result of division of two determinants.
Solutions fall to the three cases
Determinant in the denominator is not zero
then there is only solution of system
Determinant in the denominator is zero, but all numerators are not zero
then there are no solutions of system
- Determinant in the denominator is zero and all numerators are zeroes
then there are infinite set of solutions, cause one equation is linear combination of two others