Why We Use Letters: Algebraic Expressions — Class 9 Mathematics Chapter 2

🖼 Image PendingA bustling Indian neighbourhood stationery shop counter at sunset, with colourful sealed boxes of pens and pencils stacked behind the counter and a young customer reaching out for one

AI Generation Prompt

Ultra-wide cinematic banner (16:5 ratio). The interior of a small Indian neighbourhood stationery shop at warm sunset, viewed from across the wooden counter. Behind the counter are colourful stacked sealed boxes — bright red boxes of pens and deep blue boxes of pencils — neatly arranged on wooden shelves. A young schoolboy in uniform reaches across the counter, his hand resting on a single red box. The scene conveys: every shopkeeper does algebra without writing a single equation. Painterly cinematic illustration in the style of warm Indian domestic art. Dark background. No text, no labels.

✦Think About It

A shopkeeper says: *'Every red box has 4 pens, every blue box has 5 pencils.'* You buy *some* red boxes and *some* blue ones. **How would you write down a single rule for the total — without knowing how many of each you bought yet?**

What if you used *one letter* for 'how many red boxes' and *another letter* for 'how many blue ones'?

✦लीलावती — भास्कराचार्य

From Bhāskarāchārya's Lilāvatī (1150 CE) — On Practical Trade Problems

क्रयविक्रयलाभार्थं संख्यया गणितं विना

व्यापारः मूकवत् भवेत् कथं तत्र विचारणा॥

(krayavikraya-lābhārthaṃ saṃkhyayā gaṇitaṃ vinā / vyāpāraḥ mūkavat bhavet kathaṃ tatra vicāraṇā)

'खरीद-बिक्री और फायदे की बातें — अगर गणित न हो तो व्यापार गूँगा हो जाए। फिर हिसाब कैसे होगा?'

'For matters of buying, selling and profit — without numerical reasoning, trade would be mute. How then would calculation happen?'

Bhāskara II's Lilāvatī — written for his daughter — is full of trade puzzles like Raju's pen-and-pencil problem. The book taught Indian merchants algebra through everyday situations: pearls in a necklace, fruits in a basket, monkeys on a tree. Algebra was always meant to be practical. The letters stand for the numbers your shopkeeper hasn't told you yet.

Raju's shop and the magic of letters

Raju goes to a stationery shop. The shopkeeper tells him:

Every red box has 4 pens inside.
Every blue box has 5 pencils inside.

Raju decides to buy some boxes — but how many of each? Let's not commit yet. We'll just call his choices by letters:

$x$ = the number of red boxes Raju picks
$y$ = the number of blue boxes Raju picks

Now we can compute the total:

$x$ red boxes hold $4 \times x = 4x$ pens.
$y$ blue boxes hold $5 \times y = 5y$ pencils.
The shopkeeper also gives 3 free pens as a bonus.

Add them up:

$\text{Total items} = 4x + 5y + 3$

Notice what just happened. The same single expression works for every choice Raju could make. If he buys 2 red and 3 blue, total = $4(2) + 5(3) + 3 = 26$ . If he buys 5 red and 1 blue, total = $4(5) + 5(1) + 3 = 28$ . The expression $4x + 5y + 3$ doesn't depend on the specific numbers — it captures every possible answer.

This power — to write one rule that handles infinitely many cases — is the entire point of algebra.

🖼 Image PendingA close-up of two stacked sealed boxes side by side: the left is a tall red box labelled 'PENS' with several black-and-blue pens visible at the top; the right is a smaller blue box labelled 'Pencils' with sharpened yellow pencils sticking out.

AI Generation Prompt

Close-up product illustration of two sealed cardboard boxes standing side by side on a wooden surface. The left box is bright red, taller, with the word 'PENS' printed on a white horizontal label in the middle, and several black, blue, and grey pens sticking out from the open top. The right box is deep blue, slightly shorter, with 'Pencils' printed on two white labels, and sharpened yellow pencils with pink erasers sticking out. The boxes are drawn in the warm, hand-painted illustrative style of an Indian children's textbook. Dark background, orange accent labels, clean technical illustration style.

📸 Fig. 2.1 — Sealed boxes Raju sees in the shop. Red = pens, blue = pencils.

The four parts of an algebraic expression

Look at the expression $4x + 5y + 3$ again. Let's name each piece carefully — these names will come up in every chapter from now on:

Terms are the parts separated by $+$ or $-$ . In $4x + 5y + 3$ , the three terms are $4x$ , $5y$ , and $3$ .
Variables (sometimes called letter-numbers) are the letters that can change. Here the variables are $x$ and $y$ .
Coefficients are the numbers multiplying a variable. In $4x$ , the coefficient is $4$ ; in $5y$ , the coefficient is $5$ .
A constant is a stand-alone number with no variable attached. In $4x + 5y + 3$ , the constant is $3$ — the free pens that Raju got regardless of how many boxes he bought.

Quick mental check. Coefficient ≠ term. Coefficient is just the number; term is the whole piece including the variable. So in $4x$ , the coefficient is $4$ , but the term is $4x$ . And neither is the same as the variable $x$ alone. These three are different objects — get the names right and the rest of the chapter is easy.

Loading simulator…

Worked Example 1

SOLVED

For the expression $-7p + 2$ , identify: (a) the terms, (b) the variable, (c) the coefficient of $p$ , (d) the constant.

Step 1 — Spot the terms. Terms are separated by $+$ or $-$ . The expression has two terms: $-7p$ and $2$ .

Step 2 — Spot the variable. The only letter is $p$ , so the variable is $p$ .

Step 3 — Coefficient of $p$ . The number multiplying $p$ in the term $-7p$ is $-7$ . (Not $7$ — the sign belongs with the coefficient.)

Step 4 — Constant. The stand-alone number is $2$ .

Answer: terms $\{-7p, 2\}$ ; variable $p$ ; coefficient of $p$ is $-7$ ; constant $2$ .

Worked Example 2

SOLVED

For the expression $3a + 5b - 8$ , identify: (a) the number of terms, (b) the variables, (c) the coefficient of $b$ , (d) the constant.

(a) Three terms: $3a$ , $5b$ , $-8$ .

(b) Two variables: $a$ and $b$ .

(c) Coefficient of $b$ : $5$ .

(d) Constant: $-8$ . (The $-$ belongs with the constant.)

Worked Example 3

SOLVED

For the expression $\frac{1}{2}m + 9$ , identify: (a) the variable, (b) the coefficient of $m$ , (c) the constant.

(a) Variable: $m$ .

(b) Coefficient of $m$ : $\frac{1}{2}$ . Coefficients can be fractions, not just whole numbers.

(c) Constant: $9$ .

Worked Example 4

SOLVED

Evaluate the expression $4x + 5y + 3$ if Raju buys 2 red boxes and 3 blue boxes.

Step 1 — Substitute the values.

With $x = 2$ and $y = 3$ :

$4x + 5y + 3 = 4(2) + 5(3) + 3$

Step 2 — Simplify, one term at a time.

$= 8 + 15 + 3 = 26$

Answer: Raju walks home with 26 items total — 8 pens from the red boxes, 15 pencils from the blue boxes, and 3 free pens.

Logical ReasoningLevel 2 · Application

What is the difference between the expressions $4x$ and $4 + x$? Both contain a $4$ and an $x$ — are they the same?

Practice Yourself — Spot the Parts

For each expression, list the terms, variable(s), coefficients, and constant (if any). Cover the answers and check.

$7x + 11$
$-2y - 6$
$5a + 3b - 9$
$\frac{2}{3}p$
$12$
$-x + 1$
$0.5t - 4.2$
$3m + 2n + 7$

Answers: 1. terms $\{7x, 11\}$ , var $x$ , coeff $7$ , const $11$ . 2. terms $\{-2y, -6\}$ , var $y$ , coeff $-2$ , const $-6$ . 3. terms $\{5a, 3b, -9\}$ , vars $\{a, b\}$ , coeffs $\{5, 3\}$ , const $-9$ . 4. one term $\frac{2}{3}p$ , var $p$ , coeff $\frac{2}{3}$ , no constant (or constant $= 0$ ). 5. one term $12$ , no variable, the whole thing is a constant $12$ . 6. terms $\{-x, 1\}$ , var $x$ , coeff $-1$ (the unwritten 1!), const $1$ . 7. terms $\{0.5t, -4.2\}$ , var $t$ , coeff $0.5$ , const $-4.2$ . 8. terms $\{3m, 2n, 7\}$ , vars $\{m, n\}$ , coeffs $\{3, 2\}$ , const $7$ .

Quick Check

Q1.In the expression $6y - 8$ , what is the coefficient of $y$ ?

AI Generation Prompt

✦Think About It

What if you used *one letter* for 'how many red boxes' and *another letter* for 'how many blue ones'?

✦लीलावती — भास्कराचार्य

From Bhāskarāchārya's Lilāvatī (1150 CE) — On Practical Trade Problems

क्रयविक्रयलाभार्थं संख्यया गणितं विना

व्यापारः मूकवत् भवेत् कथं तत्र विचारणा॥

(krayavikraya-lābhārthaṃ saṃkhyayā gaṇitaṃ vinā / vyāpāraḥ mūkavat bhavet kathaṃ tatra vicāraṇā)

'For matters of buying, selling and profit — without numerical reasoning, trade would be mute. How then would calculation happen?'

Raju's shop and the magic of letters

Raju goes to a stationery shop. The shopkeeper tells him:

Every red box has 4 pens inside.
Every blue box has 5 pencils inside.

Raju decides to buy some boxes — but how many of each? Let's not commit yet. We'll just call his choices by letters:

$x$ = the number of red boxes Raju picks
$y$ = the number of blue boxes Raju picks

Now we can compute the total:

$x$ red boxes hold $4 \times x = 4x$ pens.
$y$ blue boxes hold $5 \times y = 5y$ pencils.
The shopkeeper also gives 3 free pens as a bonus.

Add them up:

$\text{Total items} = 4x + 5y + 3$

This power — to write one rule that handles infinitely many cases — is the entire point of algebra.

AI Generation Prompt

📸 Fig. 2.1 — Sealed boxes Raju sees in the shop. Red = pens, blue = pencils.

The four parts of an algebraic expression

Look at the expression $4x + 5y + 3$ again. Let's name each piece carefully — these names will come up in every chapter from now on:

Terms are the parts separated by $+$ or $-$ . In $4x + 5y + 3$ , the three terms are $4x$ , $5y$ , and $3$ .
Variables (sometimes called letter-numbers) are the letters that can change. Here the variables are $x$ and $y$ .
Coefficients are the numbers multiplying a variable. In $4x$ , the coefficient is $4$ ; in $5y$ , the coefficient is $5$ .
A constant is a stand-alone number with no variable attached. In $4x + 5y + 3$ , the constant is $3$ — the free pens that Raju got regardless of how many boxes he bought.

Loading simulator…

Worked Example 1

SOLVED

For the expression $-7p + 2$ , identify: (a) the terms, (b) the variable, (c) the coefficient of $p$ , (d) the constant.

Step 1 — Spot the terms. Terms are separated by $+$ or $-$ . The expression has two terms: $-7p$ and $2$ .

Step 2 — Spot the variable. The only letter is $p$ , so the variable is $p$ .

Step 3 — Coefficient of $p$ . The number multiplying $p$ in the term $-7p$ is $-7$ . (Not $7$ — the sign belongs with the coefficient.)

Step 4 — Constant. The stand-alone number is $2$ .

Answer: terms $\{-7p, 2\}$ ; variable $p$ ; coefficient of $p$ is $-7$ ; constant $2$ .

Worked Example 2

SOLVED

For the expression $3a + 5b - 8$ , identify: (a) the number of terms, (b) the variables, (c) the coefficient of $b$ , (d) the constant.

(a) Three terms: $3a$ , $5b$ , $-8$ .

(b) Two variables: $a$ and $b$ .

(c) Coefficient of $b$ : $5$ .

(d) Constant: $-8$ . (The $-$ belongs with the constant.)

Worked Example 3

SOLVED

For the expression $\frac{1}{2}m + 9$ , identify: (a) the variable, (b) the coefficient of $m$ , (c) the constant.

(a) Variable: $m$ .

(b) Coefficient of $m$ : $\frac{1}{2}$ . Coefficients can be fractions, not just whole numbers.

(c) Constant: $9$ .

Worked Example 4

SOLVED

Evaluate the expression $4x + 5y + 3$ if Raju buys 2 red boxes and 3 blue boxes.

Step 1 — Substitute the values.

With $x = 2$ and $y = 3$ :

$4x + 5y + 3 = 4(2) + 5(3) + 3$

Step 2 — Simplify, one term at a time.

$= 8 + 15 + 3 = 26$

Answer: Raju walks home with 26 items total — 8 pens from the red boxes, 15 pencils from the blue boxes, and 3 free pens.

Logical ReasoningLevel 2 · Application

What is the difference between the expressions $4x$ and $4 + x$? Both contain a $4$ and an $x$ — are they the same?

Quick Check

Q1.In the expression $6y - 8$ , what is the coefficient of $y$ ?