How do you solve #4x - 2x = -6# and #3x - 6y = -18# using matrices?
First, I am assuming you made a typo: 4x - 2y = -6 and 3x - 6y = -18?
There are several ways to solve linear systems with matrices:
- Cramer's Rule
- Inverse Matrices
- Gauss-Jordan elimination/Reduced row-echelon form
I will explain Cramer's Rule for you to try by hand, and then I will show the results of the other two methods by calculator.
For Cramer's rule, you need to be able to find the determinant of 2x2 matrices as follows:
If you notice, both denominators contain the coefficients of the original system in Standard Form.
For the numerators, replace the coefficients of the variable you seek, with the constants. That is, when solving for x (the first fraction), the 4 and 3 are replaced with -6 and -18.
The determinant of a 2x2 matrix is calculated by multiplying the first diagonal (top left to bottom right), and subtracting the product of the second diagonal from it. Try to see if you can do it yourself!
To solve by inverse matrices, do as follows:
To solve by reduced-row echelon form, do the following: