Online calculator: 3x3 Equation Solver

The user is required to enter 12 coefficients, corresponding to the coefficients of the three equations, and the calculator will then produce the solution to the system. The solution will include the values of the variables that satisfy all three equations simultaneously.

The calculator uses the following mathematical formulas for solving 3x3 system.

The system of 3 linear equations can be written as
$a_{11}x_1+a_{12}x_2+a_{13}x_3=b_1 \\ a_{21}x_1+a_{22}x_2+a_{23}x_3=b_2 \\ a_{31}x_1+a_{32}x_2+a_{33}x_3=b_3$

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

Then the solution 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 a result of a division of two determinants.

Solutions fall to the three cases.

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 one solution of system
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
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 is infinite set of solutions, cause one equation is a linear combination of two others

Solution of a system of 3 linear equations

a11

a12

a13

a21

a22

a23

a31

a32

a33

Calculation precision

Digits after the decimal point: 2

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PLANETCALC Online calculators

3x3 Equation Solver

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Solution of a system of 3 linear equations

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PLANETCALC Online calculators

Solution of a system of 3 linear equations

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