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 $z = 2x - y + 1$ : two planes crossing along the amber solution line. The figure also draws two more planes, fainter than the first two. The gray one is built from the sum of the two equations; the one drawn as an open mesh of fine lines is built 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.

Four planes crossing along one amber line. Two semi-transparent shaded planes are the original equations: one shaded from blue at its low end to red at its high end, the other from purple to yellow. A fainter gray plane is their sum, and a plane drawn as an open mesh of fine lines is twice the first minus the second; each of these different-looking combination planes still passes exactly through the same amber line. There are no sliders. — The planes of $E_1$ and $E_2$ meet along the amber solution line. The fainter gray plane is $E_1 + E_2$ and the open mesh is $2E_1 - E_2$ — each is a combination of $E_1$ and $E_2$ , and each passes through the same line.

Explore the figure

Find the two original planes first: $E_1$ , the plane $z = x + y$ , is shaded from blue at its low end to red at its high end, and $E_2$ , the plane $z = 2x - y + 1$ , from purple to yellow. Orbit until the amber line reads clearly as their intersection.
Now find the gray plane, $E_1 + E_2$ . It is tilted differently from both originals — yet watch how it threads the amber line exactly.
Look at just the pair $E_1$ and the gray plane, setting $E_2$ aside. Their intersection is the same line as before. This pair is a legal replacement for the original system.
Find the open mesh, $2E_1 - E_2$ , and check it too contains the line. Press the figure's yz view button to look straight along the $x$ -axis: the mesh shows edge-on, as a thin line, because it does not tilt in the $x$ direction at all.

Why combinations are safe

Suppose a point $(x, y, z)$ satisfies both $E_1$ and $E_2$ , written in standard form as $x + y - z = 0$ and $2x - y - z = -1$ . Then it automatically satisfies any combination: adding the equations gives

$3x - 2z = -1$

and forming twice the first minus the second gives

$3y - z = 1$

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 $E_1$ and $E_1 + E_2$ you can recover $E_2$ by subtraction, so no solutions sneak in either. Replacing $E_2$ by $E_1 + E_2$ neither loses nor gains a single point. That is the entire legality of elimination: every row operation is reversible, so the intersection — the thing we are solving for — never moves. What the three operations do to the planes differs, and only one of them can put a new plane on the page. Swapping two equations reorders the same two planes. Multiplying by a nonzero constant leaves a plane exactly where it was, since $2x + 2y = 4$ and $x + y = 2$ have identical solution sets. Adding a multiple of one equation to another is the operation that can produce a plane you did not start with — as $E_1 + E_2$ does here — though not always: adding zero times $E_1$ changes nothing, and if $E_2$ were already a multiple of $E_1$ the sum could collapse to $0 = 0$ , which is no plane at all. Reversibility is what does the work in every case; the new plane is a bonus the third operation usually brings.

Look at what the two combination planes accomplish. The plane $3x - 2z = -1$ has no $y$ in it: $y$ has been eliminated, and the plane is correspondingly slack in the $y$ direction. The plane $3y - z = 1$ 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.

Example 1 Read the solution line from the simplified planes

Use the two combination equations $3x - 2z = -1$ and $3y - z = 1$ to find a parametric description of the solution line, and confirm it matches the amber line $(t, \; (t+1)/2, \; 1.5t + 0.5)$ .

Show solution

From the first combination, $z = (3x + 1)/2$ . Substitute into the second: $3y = z + 1 = (3x + 3)/2$ , so $y = (x + 1)/2$ . Let $x = t$ . Then $y = (t + 1)/2$ and $z = (3t + 1)/2 = 1.5t + 0.5$ , which is precisely the amber line in the figure. Two simplified planes handed us the answer with almost no algebra. Pushed to completion, elimination works on the system's augmented matrix — each equation written as a row of its coefficients with its right-hand side after a bar, so that $E_1$ is the row $(1, 1, -1 \mid 0)$ and $E_2$ is $(2, -1, -1 \mid -1)$ — and reduces it to the rows $(1, 0, -\tfrac{2}{3} \mid -\tfrac{1}{3})$ and $(0, 1, -\tfrac{1}{3} \mid \tfrac{1}{3})$ : the same line, now solved for $x$ and $y$ in terms of $z$ .

\begin{bmatrix}1 & 0 & -\frac{2}{3} & -\frac{1}{3} \\ 0 & 1 & -\frac{1}{3} & \frac{1}{3}\end{bmatrix}

The system, fully reduced ✓ Computed · mojocas 0.1.0 ✓ Agrees with the text

Elimination run all the way to the end by a computer algebra system: the augmented matrix of $E_1$ and $E_2$ — the standard forms $x + y - z = 0$ and $2x - y - z = -1$ — reduced until each surviving equation is as simply oriented as a plane can be. The rows read $x - \tfrac{2}{3}z = -\tfrac{1}{3}$ and $y - \tfrac{1}{3}z = \tfrac{1}{3}$ : solve each for its leading variable and the amber line reappears, parameterized by $z$ .

An original work of XYZ Homework, built around interactive XYZ 3D figures. Chapters 1–6 follow the topic sequence of Interactive Linear Algebra (Margalit & Rabinoff, Georgia Tech, GNU FDL). Chapter 0's scope (its topics, their order, and the idea each figure shows) follows the Vectors chapter of Ximera's Linear Algebra: An Interactive Introduction. The prose, worked examples and figures were written for this book, which is not affiliated with or endorsed by the authors of either work. License: CC-BY-NC-SA-4.0.