Algebraic expressions: why we use letters

One rule, infinitely many cases

A phone plan charges a monthly fee of 15 dollars plus 2 dollars for every gigabyte of data. If you use 3 GB, you pay \(15 + 2 \cdot 3 = 21\) dollars. If you use 8 GB, you pay \(15 + 2 \cdot 8 = 31\) dollars.

The calculation changes every month, but the rule doesn't. Let \(g\) stand for the number of gigabytes, and the whole rule fits on one line:

That's the core idea of algebra: a letter can stand for a number that changes. One expression then describes every possible case at once.

Here are a few algebraic expressions:

In the last one, \(\pi\) is not a variable. It's a fixed number (about 3.14) that happens to be written with a Greek letter. Letters that stand for fixed numbers are called constants.

Writing expressions the standard way

A few conventions keep algebra compact and easy to read.

• Multiplication signs are usually left out between a number and a letter or between two letters: \(4 \cdot y\) is written \(4y\), and \(x \cdot y\) is written \(xy\).
• The number goes first: write \(7a\), not \(a7\).
• Repeated factors become powers: \(t \cdot t = t^2\) and \(t \cdot t \cdot t = t^3\).
• A coefficient of 1 is invisible: \(1x\) is just \(x\), and \(-1x\) is just \(-x\).

Between two numbers, though, you still need a symbol. Without it, \(4 \cdot 5\) would turn into \(45\).

From words to algebra

Word problems become much easier once you can translate phrases into expressions. Here is a small dictionary.

Evaluating an expression

Evaluating an expression means replacing each variable with a given number and working out the result.

When an expression can't be evaluated

Division by zero is undefined, so an expression with a variable in the denominator doesn't work for every value.

Later in the course these forbidden values come back as the excluded values of rational expressions.

Practice