Every rule you memorised in algebra is about the same one move
Flip the inequality when you multiply by a negative. Never divide by zero. Plus-or-minus when you undo a square. Do not multiply through by x. Four warnings, from four different school years.
School algebra teaches one instruction and then a list of exceptions to it. Do the same thing to both sides. Except flip the inequality when you multiply by a negative. Except never divide by zero. Except write plus-or-minus when you undo a square. Except do not multiply through by x. The exceptions arrive years apart, each with its own mnemonic, and nothing ever says what they have in common.
They have a great deal in common. There are only two moves in the whole of it, and every one of those warnings is about the second one.
Two moves
Subtracting 7 from both sides is adding −7. Dividing by 2 is multiplying by a half. Neither is a separate operation, and Negative Numbers makes the point by rewriting 5 − (−3) as 5 + 3 before it moves anything. Press ÷ there and you get the same picture as ×, because dividing by −4 is multiplying by −¼.
So the four buttons on Linear Equations are two moves wearing four labels: add something, or multiply by something. Everything from 3x + 4 = 10 up to a three-by-three system is solved with those two and nothing else.
Adding is safe, and that is the whole guarantee
Add 7 to both sides and you can subtract 7 to get back where you were. Whatever you add, whatever the equation, the move walks back and the answer you started with is still the answer.
That reversibility is what "do the same thing to both sides" is protecting. The rule is usually taught as fairness, as treating both sides alike, and symmetry is not what makes it work. A move that treats both sides identically and cannot be undone destroys the answer just as thoroughly as a move that treats them differently.
By zero, and the answer stops existing
Take x = 3 and multiply both sides by 0. You get 0 = 0, which is true. It is also true for x = 5, for x = −1, for every number there is, so the equation has stopped saying anything about x at all. Linear Equations will do it, and then tells you it is the one move on its panel that cannot be reversed.
Nothing can be walked back, because nothing is left to walk back to. Multiplication by zero is the one multiplication with no inverse, and the school warning about dividing by zero is that fact wearing a mnemonic.
By a negative, and the comparison turns round
Set −4x ≥ 12 in Inequalities and divide both sides by −4. The answer is x ≤ −3, and the sign between the two sides turned over on the way.
That comes straight off the number line. Negating reflects the line about zero, and a reflection swaps left and right: −5 is less than −3, and negating both gives 5 greater than 3. Multiplying by any negative is that same reflection, so a comparison cannot come through it pointing the same way. The move is still reversible, and what it preserves is the set of numbers that satisfy the statement rather than the direction of the sign between them.
By an unknown, and you cannot tell which case you are in
Press Multiply by x on the same tool and it will not do it. The reason is the section above: multiplying by a positive keeps the relation, multiplying by a negative reverses it, and x is the thing you have not found yet. There is no single answer to give.
This is why x/(x−2) > 1 cannot be cleared by multiplying through, and why anyone who does it gets an answer that is right on one side of 2 and wrong on the other. An equation forgives this, because an equation multiplied by an unknown gains extra solutions you can spot by substituting them back. An inequality has a region for an answer, and a region offers no short list of candidates to test one at a time.
By the number itself, and a sign is gone for good
Make r the subject of A = πr² in Rearranging Formulas and the tool prints ±√(A / π), and means it: both values, squared, give the same area. Squaring is multiplication by the number itself and it throws the sign away. No inverse can hand back information that the forward move destroyed.
Discarding the negative root is a statement about circles rather than a step of the algebra. The same shape in a physics problem often keeps both roots: a projectile at a given height has two times, and the earlier one is real even when it is not the one you wanted. The algebra cannot tell those two situations apart. A reader can.
The same two moves, all the way up
Two Equations at Once looks like a new technique and is not one. Multiply an equation through by a non-zero number, which keeps every solution because you can divide it back out. Subtract one equation from another, which keeps every solution because any pair satisfying both satisfies the difference. Choosing the multipliers so that an unknown cancels is arithmetic, not insight.
A single number decides whether there is an answer at all: a₁b₂ − a₂b₁. The tool's first sample gives −3 and one crossing, at (3, 2). Its last gives −0.02, a crossing at (−146, 300), and a warning worth more than the answer. Nudge a₁ by 0.01 and that crossing moves 108, a rate of 10,767 per unit. Nothing is wrong with the arithmetic. If those coefficients came from measurements, the crossing means almost nothing, and no step of the solving reveals it. Only the determinant does.
Three equations is Three or More at Once, and it adds no move at all. It adds an order: clear the first column, then the second, then read the answer off the bottom row and work upwards. Following it means never having to be clever about which equation to attack, which is why the same routine runs on a thousand unknowns inside every engineering package on earth.
And the last warning is the first one again
One decision survives inside that procedure: which row to divide by. Turn pivoting off, hand the tool a leading coefficient of 10⁻¹⁷ against a 1, and it dutifully divides by it. No rule is violated. x comes back as 0 where the answer is 1.
Sweep the coefficient down and watch the answer go. At 10⁻¹² it is 0.99998. At 10⁻¹⁵ it is 0.9992. At 10⁻¹⁶ it is 2.220446049250313. Those digits are machine epsilon, 2.220446049250313 × 10⁻¹⁶, the smallest number that still leaves 1 + ε different from 1: the y the tool arrives at is wrong by exactly one epsilon, and the last step divides that error by the pivot. At 10⁻¹⁷ it is zero.
The way down is not even tidy. At 10⁻¹³ the answer overshoots to 1.0003, where 10⁻¹² falls short at 0.99998 and 10⁻¹⁴ falls short again at 0.9992. The error does not even keep its sign, because what is happening in those last digits is noise rather than a trend. It stays noise right up to the point where it is the whole answer.
Multiplying by zero was forbidden in the first lesson because it erases the answer. Dividing by a number a machine cannot tell apart from zero erases it in the same way, and no rule in the school list catches that one, because it does not exist until the arithmetic is done by something with a finite number of digits. Tick the box and the algorithm divides by the largest available number instead. Every one of those cases then solves exactly, which is what the residual card on the page is for: it puts the answer back into the original equations and reports what is left over.
What the list was hiding
Four warnings, four school years, four mnemonics, and one sentence underneath all of them. Multiplication is the move that can fail to be reversible, and it fails in four different ways. By zero it destroys the answer. By a negative it keeps the answer and turns the comparison round. By an unknown it does one of those two and will not say which. By the number itself it discards a sign that no square root can give back.
Taught separately they are four things to remember. Taught together they are one question to ask about the move you are about to make, which is the question every one of them was an answer to: can I get back?
References (3)
- why the choice of pivot decides whether elimination is stable, and the growth factor that measures it Wilkinson, J. H. (1961). Error Analysis of Direct Methods of Matrix Inversion. Journal of the ACM 8(3), 281-330.
- machine epsilon, and what dividing by a quantity near it does to an answer Goldberg, D. (1991). What Every Computer Scientist Should Know About Floating-Point Arithmetic. ACM Computing Surveys 23(1), 5-48.
- partial pivoting as the standard remedy, and the residual as the test of a computed solution Higham, N. J. (2002). Accuracy and Stability of Numerical Algorithms (2nd ed.). Society for Industrial and Applied Mathematics.