`a*x + b*y = c`

Enter a value for all fields

The Cramer's Rule Calculator for Two Equations solves a two formula system of simultaneous linear equations in two variables using Cramer's Rule.

INSTRUCTIONS: Enter the following:

• (E₁)   a₁ , b₁ , c₁  (coefficients to equation1)
• (E₂)   a₂ , b_2, c₂   (coefficients to equation 2)

Solution [X,Y]: The calculator returns the values for x and y (e.g. [1,-1] respectively).

The Math / Science

Given a system of simultaneous equations:

                      a₁ • x + b₁ • y = c₁

                       a₂ • x + b₂ • y = c₂

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`

`[x, y] = [|[c_1,b_1],[c_2,b_2]| / |[a_1,b_1],[a_2,b_2]| , |[a_1,c_1],[a_2,c_2]| / |[a_1,b_1],[a_2,b_2]| ]`

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