This method consists of looking for the solution values to a system of equations by means of the determinant of a matrix. Also called **Cramer's rule**.

**Terms**

In order to apply this method, the following conditions must be met:

- The number of equations must be the same number of unknowns, that is, if we have two variables, we must have two equations.
- The determinant of the matrix of the coefficients must be non-zero.
- The equations must be prepared in such a way that the unknowns are in columns to the left of the equal sign and the independent terms to the right.

Once the above conditions are met, we can apply determinants, like this:

Be the system

a_{1}x + b_{1}y + c_{1}y = k_{1}

a_{2}x + b_{2}y + c_{2}y = k_{2}

a_{3}x + b_{3}y + c_{3}y = k_{3}

**Example**: Solve the following system of equations:

5x - 2y = - 2

- 3x + 7y = - 22

Let's see if all three conditions are met:

- We have two unknowns and two equations.
- The determinant of the coefficients is different from zero:
- The equations are prepared as required.

Considering that:

5x - 2y = - 2 → a_{1}x + b_{1}y = k_{1}

- 3x + 7y = - 22 → a_{2}x + b_{2}y = k_{2}

So:

Therefore x = - 2 and y = - 4; if we substitute them in any of the equations, we will verify that equality is fulfilled:

5 (-2) - (-4) = - 2

– 10 + 8 = – 2

– 2 = – 2

Y

– 3(-2) + 7(-4) = – 22

6 - 28 = - 22