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# Solution of a system of 3 linear equations

Solution of a system of 3 linear equations with 3 variables This content is licensed under Creative Commons Attribution/Share-Alike License 3.0 (Unported). That means you may freely redistribute or modify this content under the same license conditions and must attribute the original author by placing a hyperlink from your site to this work https://planetcalc.com/155/. Also, please do not modify any references to the original work (if any) contained in this content.

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

1. Determinant in the denominator is not zero

then there is only solution of system

2. Determinant in the denominator is zero, but all numerators are not zero

then there are no solutions of system

3. 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 #### Solution of a system of 3 linear equations

Digits after the decimal point: 2
x1

x2

x3

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