This online 3×3 System of Linear Equations Calculator solves a system of 3 linear equations with 3 unknowns using Cramer’s rule. Enter the coefficients values for each linear equation of the system in the appropriate fields of the calculator. All the fields left blank will be interpreted as coefficients with zero values. After clicking the ‘Calculate’ button you will get the values of the unknowns.
Solving a system of linear equations using Cramer’s rule
In mathematics, a system of linear equations is a set of one or more linear equations with the same number of variables (or unknowns). The linear system we consider here involves three equations with three unknowns:
$${ a }_{ 1 }x+{ b }_{ 1 }y+{ c }_{ 1 }z={ d }_{ 1 }$$ $${ a }_{ 2 }x+{ b }_{ 2 }y+{ c }_{ 2 }z={ d }_{ 2 }$$ $${ a }_{ 3 }x+{ b }_{ 3 }y+{ c }_{ 3 }z={ d }_{ 3 },$$ where \(x, y, z\) are the unknowns, \(a_1, a_2, a_3, b_1, b_2, b_3, c_1, c_2, c_3\) are the coefficients of the system, and \(d_1, d_2, d_3\) are the constant terms.
Solving a linear system of equations is a search for the values of the unknowns \(x, y, z\) such that each of the equations is satisfied. There are a number of methods for solving a system of linear equations. This linear system of equations calculator uses Cramer’s rule. It expresses the solution of the system in terms of the determinants of the coefficient matrix and of matrices obtained from it by replacing one column by the column vector of right-hand-side constant terms of the equations.
Denote by \(D\) the determinant of the coefficient matrix of the system:
$$D=\begin{vmatrix} { a }_{ 1 } & { b }_{ 1 } & { c }_{ 1 } \\ { a }_{ 2 } & { b }_{ 2 } & { c }_{ 2 } \\ { a }_{ 3 } & { b }_{ 3 } & { c }_{ 3 } \end{vmatrix}.$$
Then the determinants of the matrices obtained from the coefficient matrix by replacing one column by the column vector of right-hand-sides of the equations will be:
$$ {D_x = \left|\begin{array}{ccc} d_1 & b_1 & c_1 \\ d_2 & b_2 & c_2 \\ d_3 & b_3 & c_3 \\ \end{array}\right|,} \hspace{0.3em}
{D_y = \left|\begin{array}{ccc} a_1 & d_1 & c_1 \\ a_2 & d_2 & c_2 \\ a_3 & d_3 & c_3 \\ \end{array}\right|,} \hspace{0.3em}
{D_z = \left|\begin{array}{ccc} a_1 & b_1 & d_1 \\ a_2 & b_2 & d_2 \\ a_3 & b_3 & d_3 \\ \end{array}\right|.} $$
Cramer’s rule states that in case \(D\neq 0\) the system has a unique solution, whose individual values for the unknowns are given by the following formulas:
$$x = \frac{D_x}{D}, \hspace{0.2em} y = \frac{D_y}{D}, \hspace{0.2em} z = \frac{D_z}{D}.$$
Depending on the value of \(D\) a linear system of equations may behave in any one of three possible ways:
• If \(D\neq 0\) the system has a single unique solution, presented above.
• If \(D = 0\) and \({D_x}\neq 0\) (or \({D_y}\neq 0\) or \({D_z}\neq 0\)) the system has no solution (the linear system is inconsistent).
• If \(D = {D_x} = {D_y} = {D_z} = 0\) the system has infinitely many solutions.
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