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The ** Cramer's Rule Calculator** computes the solution and determinants for two simultaneous linear equations and three simultaneous linear equations.

The **Two Equation Solution** uses the following form:

- a
_{1}• x + b_{1}• y = c_{1} - a
_{2}• x + b_{2}• y = c_{2}

The solution utilizes the determinant of the 2x2 matrix.

The **Three Equations Solution** uses the following form:

- 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}

The solution utilizes the determinant of the 3x3 matrix.

Given a system of two simultaneous equations:

`a_1 * x + b_1 *y = c_1`

`a_2 * x + b_2 *y = c_2`

we can represent these two equation in matrix form using a coefficient matrix, as `[[a_1,b_1],[a_2,b_2]] [[x],[y]] = [[c_1],[c_2]]`, where we refer to `[[a_1,b_1],[a_2,b_2]]` as the coefficient matrix.

Using Cramer's rule we compute the determinants of the coefficient matrix: `D = |[a_1,b_1],[a_2,b_2]| = a_1*b_2 - b_1*a_2`

We also form the `D_x` and `D_y` determinants as:

`D_x = |[c_1,b_1],[c_2,b_2]|` and

`D_y = |[a_1,c_1],[a_2,c_2]|`

Continuing with Cramer's Rule, we compute the values of **x **and **y** as:

`x = D_x/D`

`y = D_y/D`

Given a system of three simultaneous equations:

`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`

we can represent these three equation in matrix form using a coefficient matrix, as `[[a_1,b_1,c_1],[a_2,b_2,c_2],[a_3,b_3,c_3]] [[x],[y],[z]] = [[d_1],[d_2],[d_3]]`, where we refer to `[[a_1,b_1,c_1],[a_2,b_2,c_2],[a_3,b_3,c_3]]` as the coefficient matrix.

Using Cramer's rule we compute the determinant of the coefficient matrix: `D = |[a_1,b_1,c_1],[a_2,b_2,c_2],[a_3,b_3,c_3]| = a_1*(b_2*c_3 - c_2*b_3) + b_1 * (c_2*a_3 - a_2*c_3) + c_1 * (a_2*b_3 - b_2* a_3)`

We also form the `D_x`, `D_y`, and `D_z` determinants as:

`D_x = |[d_1,b_1,c_1],[d_2,b_2,c_2],[d_3,b_3,c_3]|`

`D_y = |[a_1,d_1,c_1],[a_2,d_2,c_2],[a_3,d_3,c_3]|`

`D_z = |[a_1,b_1,d_1],[a_2,b_2,d_2],[a_3,b_3,d_3]|`

Continuing with Cramer's Rule, we compute the values of **x,** **y, and z** as:

`x = D_x/D`

`y = D_y/D`

`z = D_z/D`

- Cramer's Rule (two equations) [results are output as text of the form [x,y] ]
- Cramer's Rule (two equations, solved for x)
- Cramer's Rule (two equations, solved for y)
- Cramer's Rule (three equations) [results are output as text of the form [x,y,z] ]
- Cramer's Rule (three equations, solved for x)
- Cramer's Rule (three equations, solved for y)
- Cramer's Rule (three equations, solved for z)
- Determinant of 3-by-3 Matrix
- Determinant of 2-by-2 Matrix
- Illustrative video: solving system of equations with cramer's rule