Equations, inequalities and systems in Year 2

Mid-September, and in Year 2 at the Italian vocational schools (istituti professionali) the maths syllabus restarts from where it stopped in June: first-degree equations and inequalities, then linear systems. At vocational schools the written test comes every 3-4 weeks, more often than at a liceo, so the first test on these topics does not wait for November: it already falls between the end of September and mid-October.

In the first two years of vocational schools the risk of students dropping out is structurally higher than average, and maths is one of the places where it shows first: a failed subject that opens in October, if it is not closed straight away, tends to stay open all year, because the syllabus does not stop and wait. The good news is that the remedy is simple: catch up now, on a precise, small block, not in January on three months of syllabus at once.

Why the pace is different here

At vocational schools the year is almost always split into two terms, with the school report at the end of January: compared with a school with fewer tests, this means less time to lift an average that starts badly. A test every 3-4 weeks, moreover, leaves no room for “catching up at the next test”: the next test arrives before the previous one has really been sorted out, if the gap is not closed quickly. That is why, on these topics, it is best to intervene on the single block as soon as it turns out to be fragile, not wait for a more convenient moment.

What the October test really asks

The approach is applied: fewer proofs, more procedure and checking the result. The topics of these weeks:

• first-degree equations, also with brackets and fractions, as a review and consolidation of Year 1;
• first-degree inequalities, with the study of the sign to understand when the direction flips;
• linear systems of two equations in two unknowns, usually with the method the book presents first — substitution or elimination.

The critical point of these weeks is not the calculation of inequalities: it is understanding when the direction flips. That is where most marks are lost, not in the algebraic steps. (In May the INVALSI national test for the 10th grade also arrives, but before that comes the October test.)

The mistakes that cost marks

1. Dividing by a negative number without flipping the direction. From $-2x>6$, dividing by $-2$ the direction must flip:

Anyone who writes $x>-3$ did the calculation right and got only the last step wrong: the direction.

2. Distributing a minus sign in front of a bracket badly. In $5-2(x-1)$, the minus applies to both terms inside the bracket:

Writing $5-2x-2$ is the most frequent mistake at this step.

3. Multiplying only some terms of the equation by the LCM. If an equation has a fraction, the LCM of the denominators must be multiplied by every term, including those without a fraction, not just the one with the denominator.

4. Systems: substituting with a sign error. From $x=7-2y$, substituting $x$ into another equation and writing $2y$ instead of $-2y$ (or vice versa) is the most common mistake in the substitution method.

5. Not checking the solution. An equation or a system is checked by putting the result found back into the starting equation: in a test, it makes the difference between a mistake caught in time and one handed in.

6. Adding the two equations of a system for no reason. The elimination method only works if, before adding or subtracting, the coefficients of one unknown are equal or opposite. Adding $3x+2y=11$ and $2x+4y=14$ as they are gives $5x+6y=25$: a true equation, but a useless one, because it eliminates neither of the two unknowns.

Substitution or elimination: how to choose

In the applied approach of vocational schools the two main methods are these, and the choice depends on how the coefficients are written:

In exercise 3 below, the equation $2x+4y=14$ has both coefficients even and simplifies to $x+2y=7$ by dividing by 2: from there $x$ is isolated in a single step, which is why substitution is the better choice. If instead the coefficients do not simplify so easily, elimination — multiplying one equation by the right number and then adding — usually takes fewer steps.

The step-by-step method for inequalities

1. Remove the brackets, distributing with care over the signs.
2. Put the $x$ terms on the left and the numbers on the right (or vice versa), changing the sign of whatever crosses the inequality symbol.
3. Reduce to lowest terms, getting a form $ax>b$ (or $<,\ge ,\le$).
4. If $a$ is negative, divide and flip the direction. If $a$ is positive, the direction stays as it is.
5. Write the solution as an interval, not as a single number.

Three exercises worked in full

Exercise 1 — equation with brackets and a fraction. Solve:

Distribute: $3x+6-\frac{x}{4}=2x+9$. Multiply every term by 4 (the denominator):

Check: $3(4+2)-\frac{4}{4}=18-1=17$; and $2\cdot 4+9=17$. It matches.

Exercise 2 — inequality with a flipped direction. Solve:

Put the $x$ terms on the left and the numbers on the right: $3x-5x<2-8$, that is $-2x<-6$. Now divide by $-2$ and flip the direction:

Try $x=4$: $3\cdot 4+8=20$ and $5\cdot 4+2=22$, and $20<22$ is true. With $x=2$ (which should not work): $3\cdot 2+8=14$ and $5\cdot 2+2=12$, and $14<12$ is false: the inequality correctly excludes $x=2$.

To train on the same two blocks you will find the dedicated tests — first-degree equations and first-degree inequalities — organised at the pace of the science high school but on the same content you are doing: correct answer free, guided solution with your first lesson.

Exercise 3 — linear system, with a practical application. A pack of 3 notebooks and 2 pens costs €11; a pack of 2 notebooks and 4 pens costs €14. How much does a notebook cost and how much a pen?

From the second equation, dividing by 2: $x+2y=7$, so $x=7-2y$. Substitute into the first:

Then $x=7-2\cdot 2.5=2$. A notebook costs €2, a pen €2.50. Check: $3\cdot 2+2\cdot 2.5=6+5=11$; $2\cdot 2+4\cdot 2.5=4+10=14$. It matches both.

It is no accident that the test comes with a problem like this, instead of an abstract equation with $x$ and $y$ and no context: the same scheme — two unknown quantities linked by two conditions — is what you then find again in the specialist subjects, which on the commercial services track work on invoicing and stock and on the food-and-wine hospitality track on the costs and quantities of ingredients. Setting up the system is the part that counts: once it is written, the calculation is mechanical.

To practise this type of system, with instant marking, there is the Year 2 maths test — linear systems: here too, the correct answer is free, the guided solution is unlocked with your first lesson.

How we work in lessons

With a test every 3-4 weeks, catching up works best in small, immediate blocks: when it turns out in class that inequalities are not clear, the useful lesson is on that specific block, not on the whole year’s syllabus. I always start with two or three diagnostic exercises to see whether the problem is the calculation (signs, brackets, LCM) or the method (when to flip the direction, which technique to use in a system): they are two different gaps and they are closed in different ways. Waiting for the end of the term, in the first two years where the risk of losing the thread is already higher, is the choice that costs the most time afterwards — better one targeted hour straight after the first test that went badly than three hours of generic review in January.

If you haven’t had a lesson with me yet, the best way to start is the free 30-minute introductory call, with no commitment, to work out where to step in. The rates for a single lesson and the 10-hour package are on the pricing page; those who continue regularly usually move to the package.

On the same topic, but with a different audience and approach, I have written linear systems in Year 2: the four methods and when to use them — it is aimed at the liceo, with all four methods including Cramer’s rule, while here the approach stays applied. If your first Year 2 test is instead on factorising and the line, the typical technical-school programme is in factorising and the line: the first Year 2 test. The full programme of the year is on the maths page. For fractional equations and systems of inequalities at the art high school there is fractional equations and inequality systems in Year 2.

If any of these blocks is already fragile, write to me: the sooner you tackle it, the less work it takes to close it.