# Expressions

In algebra, letters are used to stand for values that can change (variables) or for values that are not known (unknowns).

A term is a number or letter on its own, or numbers and letters multiplied together, such as $$- 2$$, $$3x$$ or $$5a^2$$.

An expression is a set of terms combined using the operations +, – , x or $$\div$$, for example $$4x − 3$$ or $$x^2 – xy + 17$$.

An equation states that two expressions are equal in value, for example $$4b − 2 = 6$$.

An identity is a statement that is true no matter what values are chosen, for example $$4a \times a^2 = 4a^3$$.

## Writing expressions

### Example 1

Pens are sold in packs of 6 and rulers are sold in boxes of 10.

A teacher buys p packs of pens and r boxes of rulers. Write an expression for the total number of pens and rulers bought.

There are 6 pens in each pack, so the number of pens bought is $$6 \times p$$ which is $$6p$$.

There are 10 rulers in each box, so the number of rulers bought is $$10 \times r$$ which is $$10r$$.

The number of pens and rulers bought is $$6p + 10r$$

### Example 2

A rectangle has a width of $$x$$ cm. The height is 3 cm less than the width. Write an expression for the of the rectangle.

The perimeter is found by adding together the lengths of the sides of a shape.

The width of the rectangle is given as $$x$$ cm. The height of the rectangle is 3 less than the width: $$x - 3$$ cm

Perimeter = $$x + x + (x – 3) + (x – 3)$$

Perimeter = $$(4x - 6)$$ cm

Question

John is $$n$$ years old. Kim is three years younger than John. Vanessa is half Kim's age.

Write an expression for each person's age.

From the question, John is $$n$$ years old.

Kim is three years younger than John, so Kim is ($$n - 3$$) years old.

Vanessa is half Kim's age, so take Kim's age and divide by 2. This gives Vanessa's age as $$\frac{n - 3}{2}$$.