1.3 Row Operations Preserve the Solution

Elimination — the algorithm you use to solve systems — rewrites equations aggressively: it scales them, adds them to each other, and throws the originals away. It feels like it ought to be dangerous. Why is it legal to replace an equation with some combination of equations and still claim you are solving the same problem?

The answer is geometric, and you can watch it. The figure below starts with the system $z = x + y$ and $z = 2 x - y + 1$ : two planes crossing along the amber solution line. Hidden in the scene are two more planes. One is built from the sum of the two equations, the other from twice the first minus the second. They are genuinely different planes — different tilts, different appearance — and yet each contains the very same amber line.

Two semi-transparent planes crossing along a thick amber line. Two more planes, hidden at first, can be toggled on: each is a combination of the first two equations, and each different-looking plane still passes exactly through the same amber line. There are no sliders; the reader toggles plane visibility. — The planes of $E sub 1$ and $E sub 2$ meet along the amber solution line. Toggle the two hidden planes — each is a combination of $E sub 1$ and $E sub 2$ , and each passes through the same line.

Explore the figure

With only the two original planes visible, orbit until the amber line reads clearly as their intersection.
Reveal the plane labeled $E sub 1 plus E sub 2$ . It is tilted differently from both originals — yet watch how it threads the amber line exactly.
Hide $E sub 2$ and look at the pair $E sub 1$ with $E sub 1 plus E sub 2$ . Their intersection is the same line as before. This pair is a legal replacement for the original system.
Reveal the plane labeled $2 E sub 1 minus E sub 2$ and check it too contains the line. Try viewing it edge-on: notice it does not tilt in the $x$ direction at all.

Why combinations are safe

Suppose a point $(x, y, z)$ satisfies both $E sub 1$ and $E sub 2$ , written in standard form as $x + y - z = 0$ and $2 x - y - z = - 1$ . Then it automatically satisfies any combination: adding the equations gives

$3 x - 2 z = - 1$

and forming twice the first minus the second gives

$3 y - z = 1$

A point on both original planes has no choice but to lie on both new planes. And the reverse holds when we keep one original around: from $E sub 1$ and $E sub 1 plus E sub 2$ you can recover $E sub 2$ by subtraction, so no solutions sneak in either. Replacing $E sub 2$ by $E sub 1 plus E sub 2$ neither loses nor gains a single point. That is the entire legality of elimination: each row operation swaps one plane for a different plane through the same intersection, so the intersection — the thing we are solving for — never moves.

Look at what the two combination planes accomplish. The plane $3 x - 2 z = - 1$ has no $y$ in it: $y$ has been eliminated, and the plane is correspondingly slack in the $y$ direction. The plane $3 y - z = 1$ has no $x$ . Elimination is not just legal — each step is chosen to produce a plane so simply oriented that you can read information straight off it.

An original work of XYZ Homework, built around interactive XYZ 3D figures. Its chapter sequence is aligned to Interactive Linear Algebra (Margalit & Rabinoff, Georgia Tech, GNU FDL); this work is original, copies nothing from it, and is not affiliated with or endorsed by its authors. License: CC-BY-NC-SA-4.0.