To solve a system of linear and certain non-linear equations, the substitution method is widely used by elementary and high school students.
As explained here, to solve this simple system of linear equations,
$$ 2x – 3y = –2 $$ $$ 4x + y = 24$$
we use the first equation to isolate $y$: $y = –4x + 24$
The second equation is then substituted into the first, to solve for x (x=5), then x=5 is used in either equation to solve for y, which equals 4.
However, this method is not explicitly one of the three row operations. How would I prove the solution set obtained using this method is exactly the same as the solution set of the original system of equations?
EDIT: It has been pointed out that substitution in a linear system is in fact a row operation. But what about the use of this method for a non-linear system? For example, this method surely works for this simple non-linear system:
$$ y=x^3$$ $$ y=2x$$
(Of course, this method may not work with certain non-linear functions that are not one-to-one, say $\sin(x)$.)