Algebraic Expressions
Every question with a fully worked solution.
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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.