The Rule Hiding Inside a Growing Matchstick Pattern

A letter-number is a letter standing in for a number that can change — $n$, $x$, $a$. It lets you write one rule that covers every case at once, instead of working out each situation separately.

This page covers everything in the CBSE Class 7 Mathematics chapter's first half: writing expressions from situations, naming terms and coefficients, substituting values, and describing a growing pattern with a general expression.

Decide what the letter stands for, write it down, then build the expression from the words in order.

If one notebook costs Rs 25, then $n$ notebooks cost $25n$ rupees. Here $25n$ means $25\times n$ — the multiplication sign is left out between a number and a letter.

"Five more than a number" is $x+5$. "Three less than twice a number" is $2x-3$. "Half of a number" is $\frac{x}{2}$.

For the perimeter of a rectangle with length $l$ and breadth $b$:

$$P=2(l+b)$$

For example, if a pencil costs Rs 5 and a notebook Rs 25, then buying $p$ pencils and $n$ notebooks costs $5p+25n$ rupees — a single expression that prices any combination.

Watch the word order in subtraction. "Five less than $x$" is $x-5$, but "five less $x$" is $5-x$, and those are different numbers.

Terms are the parts separated by $+$ and $-$ signs, and the sign belongs to the term that follows it. The coefficient is the number multiplying the letter in a term.

In $7x+3y-5$:
- the terms are $7x$, $3y$ and $-5$
- the coefficient of $x$ is 7, and of $y$ is 3
- $-5$ is a constant, since it has no letter

Like terms have exactly the same letters raised to the same powers; only their coefficients differ. Unlike terms do not.

In $5a+3b-2a$, the like terms are $5a$ and $-2a$. The term $3b$ is unlike both.

Two cases to fix in your memory. In the term $x$ the coefficient is 1, because $x$ means $1\times x$. And in $-4y$ the coefficient is $-4$, sign included.

Only like terms can be combined, which is the whole reason the classification matters — and it is what the next part of this chapter is built on.

Substitute — replace each letter with the number given, then calculate. Put the substituted value in brackets so signs and powers stay correct.

Find the value of $3x+7$ when $x=4$:

$$3(4)+7=12+7=19$$

Find $5a-2b$ when $a=6$ and $b=4$:

$$5(6)-2(4)=30-8=22$$

And with a power, find $x^2+3$ when $x=5$:

$$(5{)}^2+3=25+3=28$$

Notice the power is applied before the addition. Working left to right without regard for that would give a different, wrong answer.

In real life this is what a formula does for you: $P=2(l+b)$ with $l=8$ and $b=5$ gives $2(13)=26$ — the perimeter of that particular rectangle, from the one rule that fits them all.

Count how much the pattern grows by each step, and use that as the coefficient of the position number.

Take squares made of matchsticks in a row. One square needs 4 matchsticks. Each new square shares a side with the last, so it needs only 3 more.

So the counts run 4, 7, 10, 13 — growing by 3 each time. For $n$ squares:

$$\text{matchsticks}=3n+1$$

Check it: for $n=1$, $3(1)+1=4$. For $n=4$, $3(4)+1=13$. Both match.

The 3 is the amount added each step, and the 1 is the single extra matchstick the very first square needed.

The same reasoning works on a calendar grid. Numbers in one column go down by adding 7 each time, so if the top number is $x$, the one below is $x+7$ and the next is $x+14$.

Always test your rule on at least two positions. A rule that fits the first case but not the fourth is wrong, and checking takes seconds.