Algebraic Expressions
Theory
An algebraic expression uses pronumerals to stand for numbers. This Year 9 maths page revises Year 8 algebra: turning words such as sum, product and less than into symbols, using brackets for a whole group, and substituting values, including negative numbers.
A pronumeral (or variable) is a letter that stands for a number, such as \(x\) or \(m\).
A term is a number, a pronumeral, or numbers and pronumerals multiplied together, such as \(7\), \(m\) or \(4m^2\). The number in front of a pronumeral is its coefficient.
An algebraic expression is one or more terms joined by \(+\) and \(-\) signs, such as \(4m - 7\). It has no equals sign.
Algebra has its own shorthand. Leave out the \(\times\) sign and write the number first: \(4 \times m = 4m\) and \(a \times b = ab\). Write division as a fraction: \(m \div 6 = \dfrac{m}{6}\).
To substitute (or evaluate), replace each pronumeral with its value and work out the answer.
Key words and the operations they mean:
| Words | Operation | With \(m\) and \(6\) |
|---|---|---|
| sum, plus, more than, increased by | \(+\) | \(m + 6\) |
| difference, minus, less than, decreased by | \(-\) | \(m - 6\) |
| product, times, multiplied by | \(\times\) | \(6m\) |
| quotient, divided by | \(\div\) | \(\dfrac{m}{6}\) |
| twice, double | \(\times 2\) | \(2m\) |
| half of | \(\div 2\) | \(\dfrac{m}{2}\) |
| the square of | power \(2\) | \(m^2\) |
| the cube of | power \(3\) | \(m^3\) |
How to write an expression from words
- Find the pronumerals and numbers in the phrase.
- Match each key word to its operation (see the Formulas tab).
- Build from the inside out. A phrase such as "the sum of" inside another phrase is worked out first and kept in brackets.
- Write it in shorthand: no \(\times\) sign, number first, pronumerals in alphabetical order, division as a fraction.
- Check by trying small numbers in both the words and your expression.
How to substitute
- Write the expression again with each pronumeral replaced by its value. Put negative values in brackets.
- Work out powers first, then multiply and divide, then add and subtract from left to right.
(i) seven less than the product of \(4\) and \(m\)
(ii) the square of the sum of \(a\) and \(b\).
(i) Build the product first, then take \(7\) away from it.
| \(\text{product of } 4 \text{ and } m\) | \(=\) | \(4m\) |
| \(\text{seven less than } 4m\) | \(=\) | \(\boldsymbol{4m - 7}\) |
(ii) Build the sum first. The whole sum is squared, so it needs brackets.
| \(\text{sum of } a \text{ and } b\) | \(=\) | \(a + b\) |
| \(\text{square of the sum}\) | \(=\) | \(\boldsymbol{(a + b)^2}\) |
\((a + b)^2\) is not \(a^2 + b^2\). With \(a = 1\) and \(b = 2\): \(3^2 = 9\), but \(1 + 4 = 5\).
(i) three times the difference of \(p\) and \(q\)
(ii) one third of the sum of \(x\) and \(4\).
(i) The difference is found first, in the order given. Then all of it is tripled.
| \(\text{difference}\) | \(=\) | \(p - q\) |
| \(3 \times (p - q)\) | \(=\) | \(\boldsymbol{3(p - q)}\) |
(ii) Build the sum, then divide all of it by \(3\). The fraction line keeps the sum together.
| \((x + 4) \div 3\) | \(=\) | \(\boldsymbol{\dfrac{x + 4}{3}}\) |
\(\dfrac{x}{3} + 4\) would divide only the \(x\) by \(3\).
(i) \(4p - 3q^2\) when \(p = -2\) and \(q = 3\)
(ii) \(\dfrac{ab}{a - b}\) when \(a = 6\) and \(b = -2\).
(i) Put the negative value in brackets. Square first, then multiply, then subtract.
| \(4p - 3q^2\) | \(=\) | \(4 \times (-2) - 3 \times 3^2\) |
| \(\) | \(=\) | \(-8 - 3 \times 9\) |
| \(\) | \(=\) | \(-8 - 27\) |
| \(\) | \(=\) | \(\boldsymbol{-35}\) |
(ii) Work out the top and the bottom separately, then divide.
| \(ab\) | \(=\) | \(6 \times (-2) = -12\) |
| \(a - b\) | \(=\) | \(6 - (-2) = 8\) |
| \(\dfrac{ab}{a - b}\) | \(=\) | \(\dfrac{-12}{8}\) |
| \(\) | \(=\) | \(\boldsymbol{-1.5}\) |
Subtracting a negative is the same as adding: \(6 - (-2) = 6 + 2\).
(i) the cost of \(3\) coffees and \(2\) muffins
(ii) the change from a $50 note.
All amounts are in dollars.
(i) \(3\) coffees cost \(3c\) and \(2\) muffins cost \(2m\).
| \(\text{cost}\) | \(=\) | \(\boldsymbol{3c + 2m}\) |
(ii) The change is what is left of the $50 after the whole cost is taken away.
| \(\text{change}\) | \(=\) | \(50 - (3c + 2m)\) |
| \(\) | \(=\) | \(\boldsymbol{50 - 3c - 2m}\) |
Check with \(c = 4\) and \(m = 5\): the food costs $22, so the change is $28, and \(50 - 12 - 10 = 28\).
Common pitfalls
Frequently asked questions
What is an algebraic expression?
It is a group of terms joined by \(+\) and \(-\) signs, such as \(4m - 7\). It uses pronumerals for numbers that can change. An expression has no equals sign.
What is the difference between an expression and an equation?
An expression, such as \(3x + 2\), has no equals sign and cannot be solved. An equation, such as \(3x + 2 = 11\), says two things are equal and can be solved.
What does product mean in maths?
Product means the answer to a multiplication. The product of \(4\) and \(m\) is \(4m\). Sum means add, difference means subtract and quotient means divide.
How do you write less than in algebra?
Start with the quantity that comes after the words "less than", then subtract. \(6\) less than \(m\) is \(m - 6\), not \(6 - m\).
How do you substitute a negative number?
Put the negative number in brackets, then work out powers first, then multiply and divide, then add and subtract. If \(n = -2\), then \(n^2 = (-2)^2 = 4\).