This equation solves a system of simultaneous linear equations in two variables using Cramer's Rule.

The two equations solved for here are of the form:

     `a_1 * x + b_1 *y = c_1`

     `a_2 * x + b_2 *y = c_2`

This equation returns the solution value for x.

Inputs

• `a_1` - the coefficient of the x term in the first equation
• `b_1` - the coefficient of the y term in the first equation
• `c_1` - the solution term in the first equation
• `a_2` - the coefficient of the x term in the second equation
• 'b_2` - the coefficient of the y term in the second equation
• `c_2` - the solution term in the second equation

Derivation

Given a system of 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` determinants as:

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

Continuing with Cramer's Rule, we compute the solution for x as:

     `x = D_x/D`

See also

Cramer's Rule (two equations)

Cramer's Rule (two equations, solved for y)

Cramer's Rule Calculator