Variables and constants

When you say "I bought $x$ pencils", you may mean any number , $3$, $30$ or $300$. The letter $x$ is doing the job of a flexible label.

Idea

A variable is a letter (like $x, y, a, n$) that stands for a number whose value is either unknown or can change. A constant is a fixed number like $5$ or $-3.14$.

In an expression like $3x + 7$:

• $x$ is the variable,
• $3$ is the coefficient of $x$ (the number multiplying it),
• $7$ is a constant.

Why bother with letters? Because they let us write one rule that works for many situations. The rule "double the number and add $5$" can be written as $2x + 5$. Plug in $x = 3$ to get $11$. Plug in $x = 100$ to get $205$. One expression, infinitely many answers.

Letters also let us state laws of arithmetic in a compact way. For example, the commutative law of addition is just $a + b = b + a$, true for any numbers $a$ and $b$. Try it for $a = 3, b = 7$: $3+7=7+3=10$. ✓ Try it for $a = -2, b = 5$: $-2 + 5 = 5 + (-2) = 3$. ✓

A subtle convention: in algebra we write $3x$ instead of $3 \times x$ , the multiplication sign is invisible when one factor is a letter. Similarly $xy$ means $x \times y$. But between two numbers we still need the dot or cross: $3 \cdot 4$ or $3 \times 4$.

Worked examples

Example 1. Identify the variable, coefficient and constant in $7y - 4$.

Variable: $y$. Coefficient of $y$: $7$. Constant: $-4$.

Example 2. Translate to algebra: "five more than three times a number".

Let the number be $n$. "Three times" gives $3n$. "Five more" gives $3n + 5$.

Example 3. A box has $b$ books. Three boxes have how many books?

$3 \times b = 3b$.

Example 4. State the distributive law using letters.

$a(b + c) = ab + ac$ for any numbers $a, b, c$.

Try it yourself

1. State the variable, coefficient and constant in $9 - 2k$.
2. Write "twice a number, subtract $3$" using $n$.
3. If a pencil costs $\rupee p$, how much do $8$ pencils cost?
4. State the commutative law of multiplication using letters.
5. Write an expression for "one-third of $x$, increased by $5$".
6. Identify variables and constants in $4a + 7 - 2a$.
7. Write the rule "perimeter of a square equals four times its side" using $s$ and $P$.
8. Translate: "ten less than half a number".

Activity

In pairs, make ten flashcards. One side has a sentence like "double the number plus seven"; the other side has the algebraic expression. Mix them up and quiz your partner.

Pick three different values of $x$ (your favourite numbers!) and evaluate $3x + 5$ for each. Note how only one expression handles all three.