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 perimeter of the rectangle.

Rectangle x cm long

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 \(\frac{n - 3}{2}\).