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:

• a₁ • x + b₁ • y = c₁
• a₂ • x + b₂ • y = c₂

The solution utilizes the determinant of the 2x2 matrix.

The Three Equations Solution uses the following form:

•  a₁ • x + b₁ • y + c₁ • z = d₁
•  a₂ • x + b₂ • y + c₂ • z = d₂
•  a₃ • x + b₃ • y + c₃ • z = d₃

The solution utilizes the determinant of the 3x3 matrix.

The Math

Derivation (2D)¹

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`

Derivation (3D)

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`

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Matrix Calculators

• Determinant of 3-by-3 Matrix
• Characteristic Polynomial of a 3x3 matrix
• Inverse of a 3x3 Matrix
• Transpose of a 3x3 Matrix
• Trace of a 3x3 Matrix
• Mirror of a 3x3 Matrix
• 3x3 Matrix Characteristics (Trace, Determinant, Inverse, Characteristic Polynomial)
• Product of a 3x3 matrix and a Scalar
• Product of a 3x3 matrix and a 3x1 matrix
• Product of two 3x3 matrices
• Solving 3 Equations with 3 Unknowns
• Cramer's Rule (three equations, solved for x, y and z)
• Cramer's Rule Calculator

• Illustrative video: solving system of equations with cramer's rule

1. Cramer's Rule - Wikipedia