Algebraic Expressions
An algebraic expression uses numbers, pronumerals and operations to describe a calculation, such as \(5y^3\) or \(\frac{a + 2b}{4}\). In NSW Year 8 maths you turn word descriptions into short expressions, paying close attention to order and grouping (NESA outcome MA4‑ALG‑C‑01).
You will use the words sum, difference, product and quotient, write phrases like four less than the product of 2 and \(y\) as \(2y - 4\), and show squares and cubes correctly. You will place brackets around groups, as in \(5(2x + x^2)\), and simplify the sum of three consecutive whole numbers. Writing the change Sandra gets after buying three hamburgers is a real use. It is part of Year 8 Algebra.
Every question with a fully worked solution.
- Algebraic Expressions - Video - Factorising Expressions - Example 1 Watch
Theory
An algebraic expression combines numbers and pronumerals with \(+\), \(-\), \(\times\), \(\div\) and powers, such as \(xy + 5z\). Turn words into an expression one piece at a time: the product of \(x\) and \(y\) is \(xy\); increased by five times \(z\) gives \(xy + 5z\). This Year 8 maths page shows how.
Key words. Sum means add, difference means subtract, product means multiply and quotient means divide. Increased by means add; decreased by means subtract.
Concise notation. \(3 \times m = 3m\), \(m \times n \times k = mnk\), \(k \times k = k^2\), half of \(k\) is \(\dfrac{k}{2}\), and \(p \div q = \dfrac{p}{q}\).
Order matters for subtraction and division. The difference of \(a\) and \(b\) keeps the word order: \(a - b\). But \(5\) less than \(x\) swaps it: \(x - 5\). Subtract \(5\) from \(x\) and \(5\) subtracted from \(x\) also give \(x - 5\).
Grouping words act on a whole group. Build the group first, then apply the outer operation to all of it, using brackets or a fraction line: twice the sum of \(x\) and \(5\) is \(2(x + 5)\); half the difference of \(p\) and \(3\) is \(\dfrac{p - 3}{2}\).
Powers. The square of \(k\) is \(k^2\); the cube of \(k\) is \(k^3\). A power acts only on what it sits against: \(2k^3\) cubes \(k\) alone, but \((2k)^3\) cubes the \(2\) as well.
Consecutive numbers. If \(n\) is a whole number, the next two are \(n + 1\) and \(n + 2\). Collect the result into simplest form. Money: change \(=\) money paid \(-\) total cost.
Check by putting in friendly numbers.
Same words, different groups (values when x = 6):
| Words | Expression | Value |
|---|---|---|
| twice the sum of \(x\) and \(5\) | \(2(x + 5)\) | \(22\) |
| \(5\) more than twice \(x\) | \(2x + 5\) | \(17\) |
| the square of the sum of \(x\) and \(5\) | \((x + 5)^2\) | \(121\) |
| \(5\) less than the square of \(x\) | \(x^2 - 5\) | \(31\) |
How to write an expression from words
- Find the inner group and write it first.
- Apply the outer operation to the whole group (brackets or a fraction line).
- Use concise notation and check with a number.
The product of \(x\) and \(y\) is \(xy\), and five times \(z\) is \(5z\), so the expression is \(xy + 5z\).
| \(\text{cost}\) | \(=\) | \(4b\) |
| \(\text{change}\) | \(=\) | \(a - 4b\) |
\(\$(a - 4b)\)
(i) the sum of \(d\), \(e\) and \(f\)
(ii) the product of \(6\) and \(k\), decreased by \(m\)
(iii) the quotient when \(4r\) is divided by \(s\)
(i) \(d + e + f\)
(ii) \(6k - m\)
(iii) \(\dfrac{4r}{s}\)
(i) \(8\) less than \(n\)
(ii) the difference of \(8\) and \(n\)
(iii) the product of \(3\) and \(n\), subtracted from \(20\)
(i) \(n - 8\)
(ii) \(8 - n\)
(iii) \(20 - 3n\)
With \(n = 5\), the first two give \(-3\) and \(3\): the order matters.
(i) the cube of \(2w\)
(ii) the difference between the cube of \(w\) and the square of \(w\)
| \((2w)^3\) | \(=\) | \(10^3 = 1000\) |
| \(w^3 - w^2\) | \(=\) | \(125 - 25 = 100\) |
(i) \((2w)^3\), which is \(1000\)
(ii) \(w^3 - w^2\), which is \(100\)
(ii) Ravi buys \(4\) tickets at \(\$c\) each and a \(\$6\) drink, paying with \(\$50\). Write his change.
(iii) \(n\) is a whole number. Write the sum of \(n\) and the next three whole numbers.
| \(5p + 3 \times 2p\) | \(=\) | \(11p\) |
| \(50 - (4c + 6)\) | \(=\) | \(44 - 4c\) |
| \(\text{sum}\) | \(=\) | \(4n + 6\) |
(i) \(\$11p\). Check with \(p = 20\): \(100 + 120 = 220 = 11 \times 20\).
(ii) \(\$(44 - 4c)\)
(iii) \(n + (n + 1) + (n + 2) + (n + 3) = 4n + 6\)
Common pitfalls
Frequently asked questions
What is an algebraic expression?
A combination of numbers, pronumerals and operations, with no equals sign, such as \(3x - 2\).
What does 5 less than x mean?
Start at \(x\) and take away \(5\): \(x - 5\).
What is the difference of a and b?
\(a - b\), in the order the words give.
How do you write twice the sum of x and y?
\(2(x + y)\): the brackets make the \(2\) multiply the whole sum.
How do you write consecutive whole numbers?
\(n\), \(n + 1\), \(n + 2\), and so on.