Three outcomes, and the algebra tells you which in a single row.
Two equations, two lines. Change the second equation and watch the solution appear, vanish, and become everything.
The picture and the algebra always agree.
Write the coefficients as a grid and use three moves only: swap two rows, multiply a row by a non-zero number, add a multiple of one row to another. None of them changes the solutions, which is why the method is allowed.
| Final row | Means | Picture |
|---|---|---|
| a ≠ 0 for a variable | one solution | lines cross once |
| 0 = k, k ≠ 0 | no solution | parallel, never meeting |
| 0 = 0 | infinitely many | the same line twice |
0 = 0 is not a mistake. It means the second equation told you nothing new, so the solutions form a whole line and you describe them with a parameter. 0 = 1 is not a mistake either: it means the equations contradict one another and nothing satisfies both.
The same three moves, now with planes instead of lines. x + y + z = 6, 2x − y + z = 3 and x + 2y − z = 2 meet at the single point (1, 2, 3). Substituting back into all three is the only check worth doing, and it takes fifteen seconds.
A zero determinant is the fast warning: for [[1, 1], [2, 2]] it is 1(2) − 1(2) = 0, so there is not exactly one solution. For [[2, 1], [1, −1]] it is −3, so there is.
1. Solve 2x + y = 7 and x − y = 2. Give x.
2. Find the determinant of [[1, 1], [2, 2]].
3. Row reduction ends with the row 0 = 5. The system has:
Stating which of the three outcomes you have, in words, not just stopping at a strange-looking row.
Describing an infinite solution set with a parameter rather than writing "infinitely many" and leaving it.
Substituting back into every original equation, which is the only check available once the rows have been manipulated.
These pages are free and stay free, but they are general and your IA is not. Send me your research question, or whatever exists so far, and I will tell you in writing whether the topic has a ceiling on it, where the marks are going, and what to change first. That costs nothing and it comes back within 24 hours.
Written by a serving IB Diploma and Career-related Programme Coordinator and Head of Mathematics, who reads internal assessments across every subject group every year. If you then want the whole draft reviewed properly against all five criteria, that is the paid one, and it is refunded if it does not name at least three specific things to fix.
Get a free verdict Full written review, $99
I never write any part of it. Not a sentence, not a calculation, not your data. Under 18: a parent buys this and the thread is with them. I do not work with students at my own school.