What Is Elimination Eliminating?

Thinking about a "conservation of coefficients" law in the algebraic process of elimination. How does elimination work in a purely symbolic world, where variables are never created nor destroyed? Trace the impact that vanishing coefficients have on the "simpler" equations they leave behind.

Gaussian elimination is a clever algorithm for simplifying systems of equations to make them easier to solve. When I first learned how it worked, it felt like magic.

Here is a tiny example.

Say we are given a system of two equations with two variables.

$x+2y=5$

$\mathrm{3x}+8y=19$

We can make the system easier to solve by "simplifying" the two equations. If we multiply the first equation by -3, and add it to the second equation, we get a new equation with the x variable "eliminated."

$-3\left(\mathrm{Row\; 1}\right)+\mathrm{Row\; 2}$

$-3(x+2y=5)+(\mathrm{3x}+8y=19)$

$(-3x-6y=-15)+(\mathrm{3x}+8y=19)$

$2y=14$

From here, we solve for y to get (14 / 2) = 7, then return to the first equation, plug in 2 for y, which leaves x by itself. The solution to the system is found by solving for x.

It's easy to think of that 3x as "disappearing" because, on paper at least, it vanishes from sight!

But I like to imagine that these coefficients have a sort of numerical spirit that "lives on" in the system after they disappear.

The ghost of each disappearing number actually sticks around in the equations after it is canceled out, and you can detect it... by the observable changes to the other numbers in the system via the elimination steps. The complexity of the equations does not "go away" per se... we choose to hide the complexity by scaling the other numbers proportionally to the number we are eliminating.

These number-ghosts elicit a the conservation laws from Physics, or the "there is no free lunch" rule from Economics. To make a number disappear, you have to change all of the other numbers.

Studying how the numbers change in response to the elimination steps gives us a deeper understanding of how elimination works.

Let's run another example.

Same as before, let's start with a simple system of equations, except this time, we will use symbols instead of numbers.

Because we don't know what the actual numbers are, we can't make them disappear as easily; they will leave a "trace" in the system, as we will see shortly.

This system has a lot more algebra and a lot less arithmetic:

$ax+by=c$

$dx+ey=f$

So, as before, our goal is to take the second equation and simplify it to one variable. How can we cancel out d? Well, we have to make it so that we can subtract ax from dx to get a zero.

What number can we subtract from dx to get zero? How about itself: dx.

How do we get ax to become dx? Multiply ax by (d / a).

Now before that a "disappears" on us, let's turn it red, so that we can see where it goes.

Per basic algebra rules, we must keep the whole equation unchanged by making sure every term is multiplied by (d / a) as well.

$\left(ax+by\right)\frac{d}{a}=\left(c\right)\frac{d}{a}$

Perform the multiplication, and lo and behold, the a cancels, our ax becomes dx. We are ready to subract, because ax has disappeared.

$dx+\frac{bd}{a}y=\frac{cd}{a}$

Now, we set up the subtraction, same as before: equation 2 minus equation 1...

$(dx+ey)-(dx+\frac{bd}{a}y)=f-\frac{cd}{a}$

Sure enough, the dx disappears by subtracting itself. We get 0x. The x now is gone from the second equation.

$0x+(e-\frac{bd}{a})y=f-\frac{cd}{a}$

Here is our newer, "simpler" version of equation 2. We only have y to solve for now.

$(e-\frac{bd}{a})y=f-\frac{cd}{a}$

But how "simple" does this look compared to the original example? It's true that both versions only contain one "unknown" in the original x-y sense of the word.

By replacing integers with symbols, we can now easily see that a is still there, hiding in the other numbers.

In fact, it has actually become a quite integral part of the system; it is now on both sides of the equation.

By virtue of the fractions involved, a has left us an algebraic paper trail of the transformations we did to "simplify" this system. We can preciesly trace the residue sticking to the other numbers.

The surprising hidden benefit of doing this exercise is that using pure symbols makes the history of the transformations more obvious.

By abstracting away the numerals and exploring only the relationships, we accidentally created a very usable paper trail of the operations we did.

Notice that d never really left us, either. Can you see how d lives on, hiding in the other numbers?

Elimination is less about making numbers "disappear" completely, and more about changing the relationships between the numbers in subtle ways. In other words: it's not magic... it's algebra!

I like to think of this method as another way of seeing that there is no "free lunch." Numbers in equations never vanish without consequence. You have to find ways to change all the numbers in the system to make the right relationships stand out.

This is what algebra is all about: finding interesting (or in this case, useful) relationships between numbers, and making them easier to see.