Expressions using Letter-Numbers

1. From numbers to letter-numbers

So far an expression like $7+3$ used only fixed numbers. But often we want to talk about a number that can vary or that is unknown. For "a box holds some pencils", we can call that number $x$. Then "$5$ more than the box" is simply $x+5$. The letter $x$ is a letter-number or variable — it behaves exactly like a number in calculations, we just do not know its value yet.

This small step is the doorway to algebra. It lets us state rules and relationships that hold for every number at once.

2. Writing algebraic expressions

We build expressions from variables and numbers. Note the shorthand conventions:

• $3\times x$ is written $3x$ (the multiplication sign is dropped).
• $x\times y$ is written $xy$.
• $x\div5$ is usually written $\dfrac{x}{5}$.
• $1\times x$ is just $x$ (we never write $1x$).

3. Terms, coefficients and constants

An expression is a sum of terms. In $3x+5$:

• $3x$ is a term; $3$ is its coefficient (the number multiplying the variable) and $x$ is the variable part.
• $5$ is a constant term (a fixed number, no variable).

4. Like and unlike terms

Terms with exactly the same variable part are like terms. Only like terms can be added or subtracted (combined).

• $4x$ and $9x$ are like → $4x+9x=13x$.
• $5y$ and $3y$ are like → $5y-3y=2y$.
• $4x$ and $4y$ are unlike → they cannot be combined; $4x+4y$ stays as it is.

5. Adding and subtracting expressions

To add or subtract whole expressions, collect like terms. Be careful with the signs when subtracting — the minus applies to every term being subtracted.

6. Finding the value of an expression (substitution)

Once we know what the variable equals, we substitute that number and evaluate using order of operations.

7. Using the distributive law with letters

The distributive law works the same with letter-numbers as with ordinary numbers.

8. Writing general rules and formulas

The real power of letter-numbers is stating rules that work for any value.

9. Equations: a glimpse ahead

When an expression is set equal to a number, we get an equation, e.g. $x+3=10$. The value of the variable that makes it true ($x=7$ here) is the solution. Equations are a major topic of later chapters; here we just notice they are built from the expressions we are learning to write.

The difference between an expression and an equation is worth remembering: an expression such as $2x+1$ has no equals sign and only takes a value once you choose $x$; an equation such as $2x+1=11$ makes a statement that is true for one special value of $x$. Forming the right expression first is the skill this chapter builds; solving the equation comes later.

10. Reading patterns and predicting

One of the most beautiful uses of letter-numbers in Ganita Prakash is turning a growing pattern into a single rule. The trick is to spot what stays the same (a constant start) and what grows step by step (a coefficient).

The same thinking explains why a single algebraic rule is so powerful: instead of listing terms forever, one expression like $3n-1$ captures the whole infinite pattern. This is exactly why mathematicians prefer letter-numbers — they say more, with less writing, and they let us answer "what about the $100$th?" instantly.

11. Common mistakes to avoid

• Combining unlike terms, e.g. writing $3x+4y=7xy$ — this is wrong.
• Forgetting that subtracting a bracket flips the sign of every term inside.
• Writing $1x$ instead of $x$, or $x+x$ as $x^{2}$ (it is $2x$).
• Mis-substituting: in $3x$, putting $x=4$ gives $12$, not $34$.
• Dropping the coefficient when expanding, e.g. $3(x+2)=3x+2$ (should be $3x+6$).

12. Quick revision checklist

• Variable = a letter standing for a number; $3x$ means $3\times x$.
• Coefficient = number in front; constant = lone number.
• Only like terms combine; add/subtract their coefficients.
• Substitute to find a value; follow order of operations.
• Distributive law: $a(b+c)=ab+ac$; use formulas like $P=4s$.