How do you solve using gaussian elimination or gauss-jordan elimination, #x₁+ 2x₂+ x₃= 2#, #x₁+ 3x₂- x₃ = 4#, #3x₁+ 7x₂+ x₃= 8#?
First form an augmented matrix with the coefficients on the left and the constants in the augmented part on the right. Due to formatting limitations I will use embolden font for the augmented part.
We now get the matrix in upper diagonal form using row operations:
The notation will be:
This means, row 2 is row 2 plus 3 times row 1 added to it.
Notice we have eliminated the last row. This is due to linear dependency amongst the rows. This is because one of the equations was just a multiple of the other equations. As a result we only have two independent equations, but we have 3 variables, so we can only solve the system by assigning arbitrary values to 1 of the variables. This results in there being an infinite number of solutions.
Using back substitution: