When solving a system of linear equations, you can use substitution, elimination to solve. But you also set up the system with matrices and solve using Cramer's rule.

For example, the system 2x + 3y = 10

-3x - 6y = 22

Can be solved as follows:

matrix A = 2 3

-3 -6

Get the determinant of A = (2)(-6) - (-3)(3) = -3

Now to solve for x, set up another matrix, substituting 10 for 2 in the matrix

22 -3

the new matrix looks like 10 3

22 -3

Get the determinant (10)(-3) - (22)(3) = -30 - 66 = -86

x is the determinant x divided by determinant A = -86/-3 = 86/3

Now do the same thing to get y

the new matrix is 2 10

-3 22

Determinant is (2)(22) - (-3)(10) = 44 + 30 = 74

y is 74/-3 = -74/3

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