1. Cramer’s Rule for 2x2 systems
Jerry, I'm at the corner of 1st and 1st.
Wait a minute, how can a street intersect itself?
I must be at the nexus of the universe!

2. Cramer’s Rule - 2 x 2 Cramer’s Rule relies on determinants.
Consider the system below with variables x and y:

3. Cramer’s Rule - 2 x 2
We can change this system of equations into matrix multiplication:

4. Cramer’s Rule - 2 x 2
We can solve for x in the system below by finding determinants.
The first is the determinant of the coefficient matrix: we already know how to find this:
Original matrix multiplication

5. New matrix with “x” values gone and answers instead
Cramer’s Rule - 2 x 2
Next, we need to find the determinant of a special matrix for x. We get this matrix by substituting the x values from the coefficient matrix out and the answer values in:
Original matrix multiplication
New matrix with “x” values gone and answers instead
Answers subbed in

6. Cramer’s Rule - 2 x 2
And we can find the determinant of this new, “x-swapped” matrix:
Original matrix multiplication

7. Cramer’s Rule - 2 x 2
Now we can solve for x by using the determinant of the original (coefficient) matrix and the determinant of the new, “x-swapped” matrix. This is done according to this formula:
Original matrix multiplication

8. Cramer’s Rule - 2 x 2 Example: solve for x in the following system:
We know the original matrix, we just need to find the “x-swapped” matrix:
And we are ready to solve for x by using determinants:
x = 2

9. Cramer’s Rule - 2 x 2
The same holds true for y as well as x. We just need to swap the answers in for the y-values instead. Therefore, Cramer’s Rule is summarized as follows:
Given the following matrix multiplication:
As long as the determinant of the coefficient (original) matrix does not equal zero:
Then
and
If it does equal zero, you cannot use Cramer’s Rule to solve the system.