Variables
Every question with a fully worked solution.
- Variables - Video - Algebra Multiplying & Dividing Watch
Theory
A pronumeral is a letter that stands for a number. When the letter can take different values, it is called a variable. This Year 7 maths page explains algebraic expressions and terms, how to write algebra in concise notation, how to write a division as a fraction and how to add and subtract like terms.
A car has \(4\) wheels, so \(n\) cars have \(4 \times n\) wheels. The letter \(n\) is a pronumeral. It can be \(1\), \(2\), \(10\) or any number of cars, so \(n\) is a variable.
| Cars, \(n\) | \(1\) | \(2\) | \(3\) | \(10\) |
|---|---|---|---|---|
| Wheels, \(4n\) | \(4\) | \(8\) | \(12\) | \(40\) |
An algebraic expression joins numbers and pronumerals with operations, such as \(4n\), \(x + 3\) or \(4x - 2\). Each part joined by \(+\) or \(-\) is a term.
Like terms have exactly the same pronumeral part, such as \(4m\) and \(7m\). Add or subtract the numbers in front: \(4m + 7m = 11m\). Unlike terms, such as \(4a + 5b\), cannot be joined.
Concise notation:
| Long form | Concise | Why |
|---|---|---|
| \(p \times 5\) | \(5p\) | leave out \(\times\), number first |
| \(2 \times y \times 7 \times x\) | \(14xy\) | multiply the numbers, letters in alphabetical order |
| \(1 \times x\) | \(x\) | \(1\) lot of \(x\) |
| \(b \times b\), \(c \times c \times c\) | \(b^2\), \(c^3\) | a pronumeral times itself is a power |
| \(a \div 2\) | \(\dfrac{a}{2}\) | division as a fraction |
| \(a \div (b \times c)\) | \(\dfrac{a}{bc}\) | all of \(bc\) goes underneath |
| \((c + 5) \div 2\) | \(\dfrac{c + 5}{2}\) | all of \(c + 5\) goes on top |
| \(p \div q \times r\) | \(\dfrac{pr}{q}\) | left to right: \(\dfrac{p}{q} \times r\) |
How to write an expression in concise notation
- Leave out the \(\times\) signs.
- Multiply the numbers and write the result first.
- Write the pronumerals in alphabetical order, with powers for repeats.
- Write a division as a fraction, keeping groups together.
- Collect like terms.
(i) \(y = 2\)
(ii) \(y = 8\)
(iii) \(y = 15\)
| \(20 - 2\) | \(=\) | \(18\) |
| \(20 - 8\) | \(=\) | \(12\) |
| \(20 - 15\) | \(=\) | \(5\) |
(i) \(18\)
(ii) \(12\)
(iii) \(5\)
As \(y\) gets bigger, \(20 - y\) gets smaller.
(i) \(y \times 3 - 5\)
(ii) \(4 \times b \times 7 \times a\)
(iii) \(2 \times c \times c\)
(i) Number first, leave out \(\times\). Only \(y\) is multiplied by \(3\).
(ii) Numbers first (\(4 \times 7 = 28\)), then the pronumerals in alphabetical order.
(iii) \(c \times c\) is written as a square.
(i) \(3y - 5\)
(ii) \(28ab\)
(iii) \(2c^2\)
(i) \(7 \times k + 2 \times k\)
(ii) \(8 \times w \times w - 3 \times w \times w\)
(iii) \(3 \times m + 4 \times n - 2\)
| \(7k + 2k\) | \(=\) | \((7 + 2)k = 9k\) |
| \(8w^2 - 3w^2\) | \(=\) | \((8 - 3)w^2 = 5w^2\) |
(i) \(9k\)
(ii) \(5w^2\)
(iii) \(3m + 4n - 2\): no like terms, so it cannot be simplified further. Check (i) with \(k = 2\): \(14 + 4 = 18\) and \(9 \times 2 = 18\).
(i) \(a \times a + b \times b\)
(ii) \(9 - t \times t \times 2\)
(i) \(a^2\) and \(b^2\) are unlike terms.
(ii) Number first: \(t \times t \times 2 = 2t^2\).
(i) \(a^2 + b^2\)
(ii) \(9 - 2t^2\)
(i) \((p + 6) \div 4\)
(ii) \(x \div y \times 5\)
(iii) \(10 + m \div 3\)
(iv) \(k \div (2 \times h)\)
(i) All of \(p + 6\) is divided by \(4\), so it all goes on top.
(ii) Left to right: \(\dfrac{x}{y} \times 5 = \dfrac{5x}{y}\). Check with \(x = 12\), \(y = 4\): \(12 \div 4 \times 5 = 15\) and \(\dfrac{60}{4} = 15\).
(iii) Divide before adding: only \(m\) is divided by \(3\).
(iv) All of \(2h\) goes underneath.
(i) \(\dfrac{p + 6}{4}\)
(ii) \(\dfrac{5x}{y}\)
(iii) \(10 + \dfrac{m}{3}\)
(iv) \(\dfrac{k}{2h}\)
Common pitfalls
Frequently asked questions
What is a pronumeral?
A letter that stands for a number.
What is a variable?
A pronumeral that can take different values, like the number of cars \(n\).
What are like terms?
Terms with exactly the same pronumeral part, such as \(4m\) and \(7m\).
How do you write b times b?
As a power: \(b^2\).
How do you write a divided by 2 in algebra?
As a fraction: \(\dfrac{a}{2}\).