Cramer's Rule (two equations)

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

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The Cramer's Rule Calculator for Two Equations solves a two formula system of simultaneous linear equations in two variables using Cramer's Rule. The two equations solved for here are of the form:

Formula 1 (F1): a₁ • x + b₁ • y = c₁
Formula 2 (F2): a₂ • x + b₂ • y = c₂

INSTRUCTIONS: Enter the following:

(F1) Enter a₁,b₁, c₁ separated by commas.
(F2) Enter a₂,b₂, c₂ separated by commas.

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

The Math

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:

$x = D_x/D$

$y = D_y/D$

The Math

See also