A letter in algebra is a number you have not met yet. Learn how expressions are written, how to turn words into algebra, and how to substitute numbers, including negative ones, without slipping.
A letter stands for a number. To substitute, replace each letter with its value in brackets, then follow the order of operations: brackets, then powers, then multiply and divide, then add and subtract.
Section 01
How algebra is written.
Open the lab on Substitution, pick a formula and drag the sliders for each letter. Watch the working rebuild every time: the letters never change, only the numbers sitting in their place. Then switch to Collecting like terms and press Collect to see which terms belong together.
No times sign: \(3a\) means \(3 \times a\), and \(ab\) means \(a \times b\). The number is written first: \(3a\), not \(a3\).
Division as a fraction: \(\dfrac{x}{4}\) means \(x \div 4\).
Powers: \(a^{2} = a \times a\) and \(3a^{2} = 3 \times a \times a\). Only the \(a\) is squared, not the 3.
Expression, equation, formula: \(2x + 5\) is an expression (no equals sign); \(2x + 5 = 11\) is an equation (true for one value of \(x\)); \(A = lw\) is a formula (connects quantities).
Worked example Words into algebra
Write an expression for (a) a number that is 2 more than \(n\) (b) three times \(n\), then subtract 5 (c) add 5 to \(n\), then multiply by 3.
(a) 2 more than \(n\) means add 2 to \(n\).
(b) Three times \(n\) is \(3n\); subtracting 5 gives \(3n - 5\).
(c) The adding happens first, so it must go in a bracket before multiplying by 3.
Worked example Consecutive even numbers (Extended)
\(n\) is a whole number. Write an expression for the product of two consecutive even numbers.
Any even number is 2 times a whole number, so call the first even number \(2n\).
The next even number is 2 more: \(2n + 2\).
Product means multiply.
Answer: \(2n(2n + 2)\)
Examiner trap
\(n(n + 1)\) is the product of two consecutive numbers, but not necessarily even ones. Writing \(2n\) guarantees the number is even.
Section 02
Substituting safely.
Replace every letter with its value, and put each value in brackets, especially negative numbers. Then work through the order of operations one move at a time.
Worked example Negative values
Find the value of each expression when \(a = 3\) and \(b = -2\): (a) \(4a - 3b\) (b) \(a^{2}b\) (c) \((a - b)^{2}\).
Find the value of \(2x^{2} - 5x\) when \(x = -3\).
Power first: \((-3)^{2} = 9\), so \(2x^{2} = 2 \times 9 = 18\).
\(5x = 5 \times (-3) = -15\).
\(18 - (-15) = 18 + 15 = 33\).
Answer: \(33\)
Your turn
Find the value of \(\dfrac{p^{2} - q}{2p}\) when \(p = -4\) and \(q = 6\).
\(p^{2} = (-4)^{2} = 16\), so the top is \(16 - 6 = 10\). The bottom is \(2 \times (-4) = -8\). \(\dfrac{10}{-8} = -1.25\).
Examiner trap
\(2x^{2}\) means \(2 \times x^{2}\), not \((2x)^{2}\). With \(x = -3\): \(2x^{2} = 18\) but \((2x)^{2} = 36\). And a calculator reads \(-3^{2}\) as \(-9\): type \((-3)^{2}\) with the brackets.
Stage 1 · Learn
Check your understanding
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Quick check+10 XP
Which expression means 'add 5 to a number \(n\), then multiply by 3'?
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Find the value of \(p - 2q\) when \(p = 4\) and \(q = -1\).
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Find the value of \(x^{2}\) when \(x = -3\).
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What does \(ab\) mean in algebra?
Section 03
Formulas in context.
A formula is a rule written in algebra. Write it from a description, then substitute to answer questions about it.
Worked example Writing and using a formula
A plumber charges $40 to call, plus $25 for every hour worked. (a) Write a formula for the total charge \(C\) dollars for a job lasting \(h\) hours. (b) Find the charge for a job of 3.5 hours.
(a) The hourly part is \(25 \times h = 25h\); add the fixed $40.
Use \(F = \dfrac{9C}{5} + 32\) to convert \(-15\)°C into degrees Fahrenheit.
\(\dfrac{9 \times (-15)}{5} = \dfrac{-135}{5} = -27\), then \(-27 + 32 = 5\). So \(5\)°F.
Examiner trap
Money answers need two decimal places: $127.5 should be written $127.50.
Stage 2 · Exam
Exam-style questions
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Find the value of \(3x^{2} - 2y\) when \(x = -2\) and \(y = 5\).
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Use \(E = \dfrac{1}{2}mv^{2}\) to find \(E\) when \(m = 4\) and \(v = 6\).
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Ali is \(n\) years old. His mother is 3 times as old as Ali and his sister is 4 years younger than Ali. Write an expression for the total of their three ages.
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(Extended) \(n\) is a whole number. Which expression is always the product of two consecutive even numbers?
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