Introduction to algebra.
Use letters to express a pattern, substitute values and describe a situation.
Interpret algebraic notation · substitute into expressions and formulas · form expressions
Letters stand for numbers
Suppose a pen costs $2. One pen costs $2, three pens cost $6, and any number of pens costs twice that number. A letter lets us describe this pattern without choosing the number of pens in advance.
If n is the number of pens, the cost is 2n. The missing multiplication sign is a convention: 2n means 2 × n, not the two-digit number formed by 2 and n. An expression describes a calculation; an equation makes a statement that two expressions are equal.
Algebra describes a pattern without fixing every number. A variable can take different values; an unknown is a value you need to find.
Product
3 × x. The coefficient is 3.
Power
x × x, not 2 × x.
Expression
A calculation with no equality sign.
Equation
A statement that two expressions are equal.
Formula
A relationship between quantities: area, length and width.
Pause and explainWhat does 5a mean if a is the price of one notebook?
The cost of five notebooks, found by multiplying the price a by 5.
Substitute carefully, especially negatives
Substitution means replacing a letter with a specified value. It is the bridge between a general rule and a particular numerical example. The rule stays the same; only its input changes.
Write the expression first, replace every occurrence of the letter, and use brackets around negative values. Then follow the ordinary order of operations. This makes it clear whether a negative number is being squared or a minus sign is being applied afterward.
2 × (3)² − 3 × (3) + 1 = 10
Replace every x by the same value, then follow the order of operations.
Evaluate 2a² − 3b when a = −2 and b = 5.
Square −2 inside brackets. Without brackets, −2² means −4.
Use v = u + at when u = 4, a = −3 and t = 2.
Pause and explainIf x = −3, why write x² as (−3)²?
The whole value −3 replaces x, so both negative factors are included. The square is 9.
Translate a situation into algebra
A word problem becomes easier when a letter has a precise meaning. Decide whether x represents an age, a length, a count or a price, and keep that meaning throughout the expression.
Translate one relationship at a time. “Three more than x” adds 3, while “three times x” multiplies by 3. Test your expression with a small value you understand to see whether it matches the original sentence.
If n is an integer, consecutive integers can be n, n + 1 and n + 2. Consecutive even integers can be 2n and 2n + 2.
Write the product of two consecutive even numbers.
For n = 3, this gives 6 × 8 = 48. A product needs multiplication, not addition.
Pause and explainIf a person is 4 years older than someone aged x, is their age 4x?
No. Their age is x + 4. Multiplication would mean four times the age.
Your turn. Replace every letter.
Read each expression aloud before calculating. Identify what each letter represents, where multiplication is implied, and whether the question asks for an expression or a numerical value.
For substitution, show the line with numbers replacing letters before simplifying. For word questions, try a simple numerical example to check that your algebra matches the situation.
Evaluate 3p² + 2q for p = −3 and q = −4.
Reveal worked answer
Write an expression for a taxi fare with a fixed $4 charge and $1.50 per kilometre for d kilometres.
Reveal worked answer
Write the sum of three consecutive odd integers.