# Cramer's Rule

## Cramer's Rule - How it Works - Video

### Cramer's Rule

Cramer's Rule:

The Cramer's Rule is a great tool to solve for the solution of a system of equations. Or even just one variable in a system of solutions.

If you want to solve the x variable then you will need Dx and D because x = Dx / D.

If you want to solve the y variable then you will need Dy and D because y = Dy / D.

D is the determinant of the coefficients while Dx has replaced the first column or the x coefficients with the constant terms and Dy has replaced the second column or the y coefficients with the constant terms.

### Example 1

Example 1:

In this example the determinant of the coefficients is 19. The determinant of Dx is 0 and the determinant of Dy is 2. So our final answer is the point (0, 2).

### Example 2

Example 2:

In this example the determinant of the coefficients is -7. The determinant of Dx is 4; the determinant of Dy is -5; and the determinant of Dz is -3.

So our final answer is the point (4, -5, -3).

### Can we always use it?

No. If the determinant of the coefficients is 0, then the denominator will be 0 and we can't have that. The solution could be no solution (inconsistent) or an infinite solution (dependent), for example, (a, a+1, a+2).