### Elementary Row Operations

(Interchange) Interchange two equations
(Scaling) Multiply an equation by a non-zero constant
(Replacement) Replace one row by the sum of itself and a multiple of
another row [like r2'= -3r1 + r2]
### Systematic Method to achieve Gaussian Elimination

Save the x term in equation 1 and use it to eliminate all the other x terms below it via r_{k}' = c r_{1} + r_{k}
Ignore equation 1 and use the y term in equation 2 to eliminate all the
y terms below it.
Continue until the matrix is in *Gaussian* or echelon form,
with 0s below the diagonal (interchange rows as needed)
If the system is *consistent* then the
last row with non-zero coeffients will yield x_{k}=b, and then
back substitution can be used to solve for the variables.
### Continuing to Gauss-Jordan/ReducedRowEchelon form

Scale the last row with non-zero coefficients
so that the diagonal entry is a 1.
Use the last non-zero equation to eliminate the spots *above* it
Repeate these steps using the second last non-zero equation.
Continue until the matrix is in *Gauss-Jordan/ReducedRowEchelon* form
with 0s or 1s on the diagonal and 0 coeffients everywhere else.
Read off the solutions