Algebraic expressions

An algebraic expression is notation where letters stand beside numbers: $3x + 5$, $2(a - b)$, $\frac{x}{2} - 1$. A letter stands for a number — one we do not know yet, or one that is allowed to change. That way a single expression describes a whole family of situations instead of one case at a time.

A ticket costs 12 units and a seat reservation 5. The cost of buying $x$ tickets with one reservation is:

This one line answers the question for any number of tickets — just substitute a number for $x$.

Term, coefficient, variable

An expression is built from terms — the parts separated by plus and minus signs. In $12x + 5$ the terms are $12x$ and $5$.

• variable — the letter standing for a number ($x$),
• coefficient — the number in front of the variable ($12$),
• constant term — a term with no variable ($5$).

Writing $12x$ means multiplication: $12 \cdot x$. The multiplication sign between a number and a letter is dropped, because nothing else could be meant.

Like terms and how to collect them

Like terms are terms with the same letter raised to the same power. $3x$ and $7x$ are like terms; $3x$ and $3x^2$ are not, and neither are $3x$ and $3y$.

Like terms can be merged into one — that is collecting like terms. Only the coefficients are added; the letter stays as it is:

This is not a new rule but the distributive law read right to left: $3x + 2x = (3 + 2)x = 5x$. Three apples and two apples make five apples — not "five apples squared".

Collecting terms does not "work anything out". It writes the same object more briefly — and a shorter expression is easier to evaluate and harder to get wrong.

The distributive law

Multiplying a number by a sum means multiplying it by every term:

The area of a rectangle shows why: a rectangle with sides $a$ and $b + c$ can be cut into two rectangles of areas $ab$ and $ac$. The same rule holds for subtraction: $a(b - c) = ab - ac$.

A minus in front of a bracket

A minus in front of a bracket is multiplication by $-1$, so every term inside flips its sign — including one that was already negative:

This is where a sign gets lost most often. When a bracket is being subtracted, expand first and collect terms afterwards.

The value of an expression

To evaluate an expression, substitute a number for the letter and carry out the operations in the usual order. For $8x + 6$ with $x = 4$:

Simplifying an expression never changes its value — $5x + 8 + 3x - 2$ with $x = 4$ also gives $38$. That is the simplest way to check a rearrangement: substitute the same number into both forms and compare.

The exercises below ask for an expression. What counts is its value, not how you write it: $5x + 10$, $10 + 5x$ and $5 \cdot x + 10$ are one and the same answer. Copying the question back is not — the point is the simpler form.

Common mistakes

• Adding unlike terms — $3x + 4$ is already the answer; there is no "$7x$" to be had.
• Confusing adding with multiplying letters — $x + x = 2x$, but $x \cdot x = x^2$.
• Losing a sign after a minus in front of a bracket — $-(x - 3)$ is $-x + 3$, not $-x - 3$.
• Expanding only the first term — in $3(2x + 4)$ the three multiplies both terms, so $6x + 12$, not $6x + 4$.