A3: Algebraic Terminology
Understand expressions, equations, formulae, identities, inequalities, terms, factors
π Key Concepts
Understanding terminology is essential: Different algebraic forms have different meanings and purposes.
Key Terms
| Term | Definition | Example |
|---|---|---|
| Expression | Collection of terms, no equals sign | 3x + 2, 5aΒ² - b |
| Equation | Statement that two expressions are equal | 3x + 2 = 14 |
| Formula | Rule showing relationship between variables | A = ΟrΒ², v = u + at |
| Identity | Always true for all values (β‘) | 2(x + 3) β‘ 2x + 6 |
| Inequality | Shows one value is less/greater than another | x > 5, y β€ 10 |
π Expressions
Expression: A combination of numbers, variables and operations. No equals sign.
Example 1
Identify the parts of the expression 5xΒ² + 3x - 7:
Terms: 5xΒ², 3x, -7 (separated by + and -)
Coefficients: 5 (of xΒ²), 3 (of x)
Constant: -7
Variables: x
Example 2
How many terms are in each expression?
a) 4a + 2b - 3 β 3 terms
b) xΒ²y β 1 term
c) 2(x + 3) β 1 term (everything inside brackets is one term)
π Equations
Equation: States that two expressions are equal. Can be solved to find unknown values.
Example 3
Which of these are equations?
a) 3x + 2 = 11 β Yes (can solve: x = 3)
b) 5x - 3 β No (this is an expression)
c) y = 2x + 1 β Yes (this shows a relationship)
Solving equations: Find the value(s) of the variable that make the equation true.
π Formulae
Formula: A mathematical rule written as an equation. Shows how to calculate one quantity from others.
Example 4
Common formulae:
a) Area of rectangle: A = lw
b) Area of circle: A = ΟrΒ²
c) Speed: s = dβt
d) Pythagoras: aΒ² + bΒ² = cΒ²
Example 5
Using the formula A = lw, find A when l = 8 and w = 5:
A = 8 Γ 5 = 40
π Identities
Identity: An equation that is always true for ALL values. Uses the symbol β‘ (identically equal to).
Example 6
Which are identities?
a) 2(x + 3) β‘ 2x + 6 β Yes (true for all x)
b) x + 5 = 10 β No (only true when x = 5)
c) (a + b)Β² β‘ aΒ² + 2ab + bΒ² β Yes (true for all a and b)
Testing: To check if something is an identity, try different values. If it works for all values, it's an identity.
π Inequalities
Inequality: Shows that one expression is greater than or less than another.
| Symbol | Meaning |
|---|---|
| < | Less than |
| > | Greater than |
| β€ | Less than or equal to |
| β₯ | Greater than or equal to |
Example 7
x > 3 means x can be any number greater than 3 (3.1, 4, 100, etc.)
y β€ 10 means y can be 10 or any number less than 10
π Terms and Factors
Term: Part of an expression separated by + or - signs.
Factor: Numbers or variables that multiply together to make a term.
Factor: Numbers or variables that multiply together to make a term.
Example 8
Expression: 5xΒ² + 3xy - 7
Terms: 5xΒ², 3xy, -7
Factors of 5xΒ²: 5, x, x (or 5 and xΒ²)
Factors of 3xy: 3, x, y
Example 9
Find the common factors in 6x and 4xΒ²:
6x = 2 Γ 3 Γ x
4xΒ² = 2 Γ 2 Γ x Γ x
Common factors: 2 and x
Highest common factor: 2x
β Practice Questions
Q1: Is 4x + 7 an expression, equation, formula or identity?
Q2: How many terms are in the expression 3aΒ² - 2ab + 5b - 1?
Q3: Is 2(x + 4) = 2x + 8 an identity? Explain why.
Q4: List the factors of the term 12xy.
Q5: What does x β€ 8 mean?
Q6: Write an equation using x that has solution x = 4.
β Answers
- Expression (no equals sign)
- 4 terms
- Yes, it is an identity because it's true for all values of x
- Factors: 1, 2, 3, 4, 6, 12, x, y, 2x, 3x, 4x, 6x, 12x, xy, 2xy, 3xy, 4xy, 6xy, 12xy (and various combinations)
- x is less than or equal to 8 (x can be 8 or any smaller number)
- Examples: x + 2 = 6, 2x = 8, x - 4 = 0, etc.
π― Exam Tips
- Expressions have NO equals sign
- Equations can be solved (have specific solutions)
- Identities are always true (for all values)
- Formulae show relationships between quantities
- Terms are separated by + and - signs
- Factors multiply together within a term
π§ Problem-Solving Strategies
Problem-Solving
Classifying algebraic statements correctly is essential. Check: does it have an equals sign? Is it always true or only for specific values? Can you solve it? These questions determine whether it is an expression, equation, formula, identity or inequality.
Multi-Step Problem
Sam writes 2(x + 4) = 2x + 8. Mia writes 2(x + 4) = 10. (a) Classify each statement. (b) For the equation, find x. (c) How could Sam prove their statement is an identity?
Solution:
(a) Sam's is an identity (true for ALL x). Mia's is an equation (true only for x = 1).
(b) 2x + 8 = 10 β 2x = 2 β x = 1
(c) Expand LHS: 2(x + 4) = 2x + 8 = RHS. Or test multiple values β always works. Use β‘ symbol.
β οΈ Common Errors
Watch Out!
1. Wrong: 2x + 5 is an equation Correct: 2x + 5 is an expression (no equals sign)
2. Wrong: An identity is true for some values Correct: An identity is true for ALL values β use β‘ not =
3. Wrong: "x" has coefficient 0 Correct: "x" has coefficient 1 (since x = 1x)
βοΈ 6-Mark Exam Question
Extended Answer
6 marks: For each statement below, state whether it is an expression, equation, formula, identity or inequality. Give a reason for each. (i) 4a - 3 (ii) v = u + at (iii) 3(x - 1) β‘ 3x - 3 (iv) 2x + 5 > 11 (v) 5x - 7 = 13
(i) Expression β no equals sign, just terms combined with operations.
(ii) Formula β shows relationship between variables (velocity, initial velocity, acceleration, time).
(iii) Identity β true for ALL values of x (expand LHS: 3x - 3 = RHS). Uses β‘.
(iv) Inequality β uses > symbol, shows x is greater than a value.
(v) Equation β can be solved: x = 4.
Mark scheme: 1 mark each for correct classification, ΒΌ mark each for reason (6 total).
π AO3: Reason & Interpret
Reasoning and Interpretation
A teacher writes: "The perimeter of a square is P = 4s."
(a) Is this an equation, formula or identity? Explain.
(b) Rearrange to make s the subject.
(c) A student says "P = 4s is an identity because it's always true." Is this correct?
Answers: (a) Formula β it shows how to calculate P from s. (b) s = Pβ4. (c) No β it's a formula, not an identity. An identity must be true for ALL values on both sides, but P = 4s only relates specific quantities. If s = 3 then P = 12, but you cannot substitute P = 50 into the left and right independently.
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