Chapter 4: Expressions Using Letter-Numbers

So far, every number you wrote was a fixed value like $5$ or $42$. Now imagine a number that can change. Sometimes it stands for the age of a person, sometimes for the cost of a pencil, sometimes for an unknown that you must find. Mathematicians use letters like $x$, $y$ or $a$ for such numbers.

A letter-number (or variable) is just a placeholder for a value. Once you decide what value it takes, the whole expression has a specific meaning. For example, $3x + 5$ becomes $3(2) + 5 = 11$ when $x = 2$.

In this chapter you will learn to read and write algebraic expressions, evaluate them for given values, combine like terms, and solve simple equations like $x + 4 = 9$. These are your first steps into algebra , one of the most powerful languages in mathematics.

Algebra is the reason a single formula can describe thousands of situations. The expression $d = v \times t$ works for a snail, a train, and a rocket , only the values of $v$ and $t$ change. By the end of this chapter you will be comfortable working with such universal expressions.

What's inside

1. Variables and constants , the difference between a fixed number and a placeholder.
2. Reading and writing expressions , turning sentences into algebra.
3. Evaluating expressions , substituting values for letters.
4. Like terms and simplification , adding apples with apples.
5. Simple equations , finding the value of the unknown.

Key results

Useful identities you will use throughout the chapter: $x + x = 2x, \quad 3x + 5x = 8x, \quad a(b+c) = ab + ac.$

How to read this chapter

Whenever you see a letter, try substituting small numbers like $0, 1, 2$ to see what the expression evaluates to. Algebra becomes intuitive when you constantly check it against concrete numbers.