the basic idea is this:

if:

A+B = C

and

D+E = F

then C+F = (A+B) + (D+E) (since A+B is just C, and D+E is just F).

now let's put x's and y's in this:

ax + by = c

dx + ey = f

(here, a,b,c,d,e and f are numbers, and x and y are variables we want to find values for).

so (a+d)x + (b+e)y = c+f

now if it just so happens that a = -d, then a + d will be -d + d = 0, so that term will "disappear". but if not, we can MAKE that happen:

d(ax + by) = cd (since the two sides were equal, they're still equal if we multiply both sides by the same number, d).

(-a)(dx + ey) = -af

that is:

(ad)x + (bd)y = cd

(-ad)x + (-ae)y = -af

and adding these two equations, we get:

(ad - ad)x + (bd - ae)y = cd - af

that is:

(bd - ae)y = cd - af.

if bd - ae isn't 0 (it might be), we can divide by it, and get:

y = (cd - af)/(bd - ae), and then find x by substitution (it turns out that x = (ce - bf)/(ae - bd)).

the point being, none of the things i did with the equations above depended on any special properties of x or y (we made no assumptions about what they might be), so if x and y satisfy the original equations:

ax + by = c

dx + ey = f

then they still satisfy the rearranged equations:

x = (ce - bf)/(ae - bd)

y = (cd - af)/(bd - ae) <---provided the denominator isn't 0 (if it is, we have a "degenerate" system of equations. it turns out that if we express this in matrix terms, we have a non-invertible matrix).

you are quite correct, that all row operations do is mimic (or as mathematicians like to say: formalize) the process you learned way back when, of "eliminating variables".