What the solver is actually finding, and the case it cannot find because it is not there.
Two equations, two lines. The solution of the system is the single point on both of them. Tilt the second line and watch the solution move, then keep going until there is none.
Both equations are satisfied at exactly one point.
Always substitute back. It takes ten seconds, it catches a mistyped coefficient, and it is the only check you have when the method was "I typed it into the calculator".
Three variables works the same way, with planes instead of lines. For x + y + z = 6, 2x − y + z = 3 and x + 2y − z = 2 the solution is (1, 2, 3), the one point lying on all three planes.
In examinations there will always be a unique solution. That is a promise about the question, not about mathematics: parallel lines give none, identical lines give infinitely many, and real problems do both. Knowing the picture is what lets you say which.
Roots, zeros and x-intercepts are the same thing seen from three directions: the solutions of f(x) = 0, the inputs that make f zero, and where the graph meets the x axis. Questions switch between the three words freely and expect you to follow.
Graph it before you trust the solver. A cubic has at most three real roots and may have only one. The graph tells you how many to look for, so you know whether the calculator has given you all of them or just the one nearest your guess.
A cafe sells three sizes. Two small and one large cost 155 baht. One small and three medium cost 190 baht. One medium and two large cost 200 baht.
Find the price of each size.
1. Solve 2x + y = 7 and x − y = 2. Give x.
2. x³ − 6x² + 11x − 6 = 0. Give the largest root.
3. Two lines in a system turn out to be parallel and distinct. The system has:
Writing the equations down in a consistent form before entering anything. Most lost marks here are typing errors, not method errors.
Giving every root when the question says "solve", not just the first one the calculator offers.
Substituting back, and saying what the solution means in context when the question came from a real situation.
These three are deliberately not all about this page. In an examination the hardest step is deciding which method applies, and questions that arrive straight after the method never make you decide. Answer from memory before you reveal anything.
Come back to these in a week, and again in a month.
Which method?
Two lines in a system are parallel and distinct. How many solutions?
Which method?
Find 16 to the power minus a half.
Which method?
Eigenvalues of a 2×2 matrix are 3 and 1. What is its determinant?
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.