Is there any trick to solve system of linear equations by Row Equivalent Method?

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Whenever I try to solve three equations (having x, y and z) variables, I never get to the solution. Sometimes, complex fractions are formed during the process, sometimes large numbers and even if I make the matrix a unit matrix, the answers are incorrect... Is there any way (trick) to get the notion of the correct steps or somehow get to the correct answer (without having to think for long)?