Two online calculators and solvers for systems of 2 by 2 and 3 by 3 linear equations using Cramer's rule and showing the steps.

\[ x = \dfrac{D_x}{D} , y = \dfrac{D_y}{D} \]

where the determinant of a 2 by 2 Matrix, \( D \), \( D_x \) and \( D_y \) are defined by

\( D = \begin{vmatrix}a_1&b_1\\ a_2&b_2\end{vmatrix} = a_1 b_2 - b_1 a_2\)

\( D_x = \begin{vmatrix}\color{red}{c_1} & b_1\\ \color{red}{c_2} & b_2\end{vmatrix} = c_1 b_2 - b_1 c_2\)

\( D_y = \begin{vmatrix}a_1 & \color{red}{c_1}\\ a_2 & \color{red}{c_2}\end{vmatrix} = a_1 c_2 - c_1 a_2\)

Camer's rules give a solution for \( D \ne 0\).

This tool can be used to check the solutions of a 2 by 2 system of equations solved by hand. It can also be used, efficiently, to explore 2 by 2 system of equations using different values for the coefficients.

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\[ \left\{ \begin{array}{lcl} a_1 x + b_1 y + c_1 z = & \color{red}{d_1}\\ a_2 x + b_2 y + c_2 = & \color{red}{d_2} \\ a_3 x + b_3 y + c_3 = & \color{red}{d_3} \\ \end{array} \right. \] and the solutions are given by

\[ x = \dfrac{D_x}{D} , y = \dfrac{D_y}{D} , z = \dfrac{D_z}{D} \]

for \( D \ne 0 \) and where \( D, D_x, D_y \text{and} D_z \) are determinants of 3 by 3 matrices defined by

\( D = \begin{vmatrix} a_1&b_1&c_1\\ a_2&b_2&c_2 \\a_3 & b_3 & c_3 \end{vmatrix} \)

\( D_x = \begin{vmatrix}\color{red}{d_1} & b_1 & c_1\\ \color{red}{d_2} & b_2 & c_2 \\ \color{red}{d_3} &b_3&c_3 \end{vmatrix} \), \( D_y = \begin{vmatrix}a_1&\color{red}{d_1}&c_1\\ a_2&\color{red}{d_2}&c_2\\a_3 & \color{red}{d_3} & c_3 \end{vmatrix}\) , \( D_z = \begin{vmatrix}a_1&b_1&\color{red}{d_1}\\ a_2&b_2&\color{red}{d_2}\\a_3 & b_3 & \color{red}{d_3} \end{vmatrix}\)

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Cramer's Rule to Solve Systems of Equations.

Solve Systems of Equations.

Maths Calculators and Solvers.