Free · Year 2 of high school · 50 minutes · 15 questions · no penalty

Practice maths class test — Year 2: linear systems.

15 questions drawn at random from a bank of 25, on the block of the programme your teacher is finishing: every attempt is a different test. No penalty for wrong answers, so answer even if you are not sure. At the end you receive by email the correct answer to every question and your weak topics — with an account you see them straight away here on the page. The step-by-step solution unlocks with your first lesson.

A class test is the written test an Italian high-school class sits after each block of the programme; this practice version is multiple choice.

Test in ItalianThe questions of this simulator are in Italian, as in the real exam.

Test prepared by Gaetano Livornese · Aerospace engineer, Politecnico di Milano · maths and physics tutor · updated 13 October 2026

Programme covered by the test

The substitution method
The comparison method
The elimination method
Cramer’s rule
Systems with one solution, infinitely many or none
Problems solved with a system

Two worked examples from the test

Question 1 · Sistemi lineari (sostituzione, confronto, riduzione, Cramer) · Sistemi lineari

Risolvi con il metodo di sostituzione:

${x = 3 y - 2 2 x + y = 10$

A $(2, 4)$
B $(4, - 2)$
C $(4, 2)$ correct answer
D $(7, 3)$

Solution

La prima equazione dà già $x$ in funzione di $y$ : sostituiscila nella seconda.
$2 (3 y - 2) + y = 10$
Svolgi la parentesi: $6 y - 4 + y = 10$ , cioè $7 y = 14$ .
Quindi $y = 2$ .
Torna alla prima: $x = 3 \cdot 2 - 2 = 4$ .
La soluzione è la coppia $(4, 2)$ : verifica nella seconda, $2 \cdot 4 + 2 = 10$ . Corretto.

Question 2 · Sistemi lineari (sostituzione, confronto, riduzione, Cramer) · Sistemi lineari

Risolvi con il metodo di riduzione:

${3 x + 4 y = 18 5 x - 2 y = 4$

A $(2, 3)$ correct answer
B $(3, 2)$
C $(2, - 3)$
D $(- 2, 6)$

Solution

I coefficienti di $y$ non sono opposti: moltiplica tutta la seconda equazione per $2$ .
Ottieni $10 x - 4 y = 8$ (anche il termine noto va moltiplicato).
Somma con la prima: $13 x = 26$ , quindi $x = 2$ .
Sostituisci nella prima: $6 + 4 y = 18$ , cioè $4 y = 12$ e $y = 3$ .
Soluzione $(2, 3)$ ; controllo: $5 \cdot 2 - 2 \cdot 3 = 4$ . Corretto.

In the online practice test you will find 13 more questions like these; the correct answer to every question arrives by email.

Prefer paper? Download the test as a PDF (15 questions, 4 pages · 84 KB, ready to print). You receive the solutions by email after taking it online.

Download the PDF

What a maths test on linear systems looks like

The same system solved with two different methods to show that the result matches, a system with fractions to reduce first, a system to recognise as having infinitely many solutions or none, and at least one problem to turn into two equations.

The mistake that costs the grade

Concluding that a system has no solution when it has infinitely many, or the other way round: 0 = 0 means infinitely many solutions, 0 = 5 means none. In problems, the costliest mistake is setting up a single equation with two unknowns and noticing ten minutes later.

How to use this practice test

Take it once cold, without revising and with the timer on: it is there to show you where you stand, not to look good. Then look at your weak topics, revise only those — usually two or three, not the whole chapter — and retake it after a few days. The 15 questions are drawn from a bank of 25, so the second attempt is not the same test and the scores are genuinely comparable.

Is the score not enough?

See which topics lost you the most marks, then read how to handle test anxiety — or we look at your results together: the diagnostic is free.

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