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# Solution of a system of 3 linear equations

Solution of a system of 3 linear equations with 3 variables

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There are common formulae for solutions of system of 3 linear equations as
$\left{a_{11}x_1+a_{12}x_2+a_{13}x_3=b_1\\a_{21}x_1+a_{22}x_2+a_{23}x_3=b_2\\a_{31}x_1+a_{32}x_2+a_{33}x_3=b_3$

They are usually expressed using determinant of 3x3 matrixes, which is defined as
$\left|\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix} \right| = a_{11}a_{22}a_{33} + a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32} - a_{13}a_{22}a_{31} - a_{11}a_{23}a_{32} - a_{12}a_{21}a_{33}$

The solutions then will be
$x=\frac{\left|\begin{matrix} b_1 & a_{12} & a_{13} \\ b_2 & a_{22} & a_{23} \\ b_3 & a_{32} & a_{33} \end{matrix} \right|}{\left|\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix} \right|}\\y=\frac{\left|\begin{matrix} a_{11} & b_1& a_{13} \\ a_{21} & b_2 & a_{23} \\ a_{31} & b_3 & a_{33} \end{matrix} \right|}{\left|\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix} \right|}\\z=\frac{\left|\begin{matrix} a_{11} & a_{12} & b_1\\ a_{21} & a_{22} & b_2 \\ a_{31} & a_{32} & b_3 \end{matrix} \right|}{\left|\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix} \right|}$
thus each variable is result of division of two determinants.

Solutions fall to the three cases

1. Determinant in the denominator is not zero
$\left|\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix} \right|<>0$
then there is only solution of system

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

$\left|\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix} \right|=0\\\left|\begin{matrix} b_1& a_{12} & a_{13} \\ b_2 & a_{22} & a_{23} \\ b_3 & a_{32} & a_{33} \end{matrix} \right|<>0\\\left|\begin{matrix} a_{11} & b_1& a_{13} \\ a_{21} & b_2 & a_{23} \\ a_{31} & b_3 & a_{33} \end{matrix} \right|<>0\\\left|\begin{matrix} a_{11} & a_{12} & b_1\\ a_{21} & a_{22} & b_2 \\ a_{31} & a_{32} & b_3 \end{matrix} \right|<>0$

then there are no solutions of system

3. Determinant in the denominator is zero and all numerators are zeroes $\left|\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix} \right|=0\\\left|\begin{matrix} b_1& a_{12} & a_{13} \\ b_2 & a_{22} & a_{23} \\ b_3 & a_{32} & a_{33} \end{matrix} \right|=0\\\left|\begin{matrix} a_{11} & b_1& a_{13} \\ a_{21} & b_2 & a_{23} \\ a_{31} & b_3 & a_{33} \end{matrix} \right|=0\\\left|\begin{matrix} a_{11} & a_{12} & b_1\\ a_{21} & a_{22} & b_2 \\ a_{31} & a_{32} & b_3 \end{matrix} \right|=0$

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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