3x3 Equation Solver

This calculator is designed to solve systems of three linear equations.

Timur

Created: 2011-06-16 22:02:18, Last updated: 2023-02-10 11:44:51

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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.

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 one 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 is infinite set of solutions, cause one equation is a 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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PLANETCALC, 3x3 Equation Solver