Equations and Identities
This topic covers the crucial difference between equations, which are true for specific values, and identities, which are true for all values. Mastering this allows you to confidently manipulate and simplify algebraic expressions, a core skill for solving more complex problems.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- An equation is a statement that is true for a finite number of 'solutions'. For example, x2 = 4 is only true for x = 2 and x = -2.
- An identity is a statement that is true for all possible values of the variables. For example, (x+y)2 ≡ x2 + 2xy + y2.
- The symbol '≡' explicitly means 'is identically equal to', though '=' is often used for both identities and equations.
- To prove two expressions are equivalent (i.e., that they form an identity), you must manipulate one side algebraically until it is identical in form to the other side.
- Do not 'solve' a potential identity by moving terms across the equals sign. Work on the Left-Hand Side (LHS) and Right-Hand Side (RHS) independently.
Formulae
a2 - b2 ≡ (a - b)(a + b) To factorise a difference of two squares, or to expand the product of a sum and a difference.
(a + b)2 ≡ a2 + 2ab + b2 To expand the square of a sum. Be careful not to mistake it for a2 + b2.
(a - b)2 ≡ a2 - 2ab + b2 To expand the square of a difference. Watch for the sign of the middle term.
Definitions
- Equation
- A mathematical statement asserting that two quantities are equal, which is only true for a specific set of values for its variables.
- Identity
- An equation that holds true for all possible values of its variables. Attempting to 'solve' it typically results in a trivial statement like 0 = 0.
- Equivalent Expressions
- Two algebraic expressions that have the same value for all variable substitutions. Proving equivalence is the same as proving an identity.
Worked example
Determine if the following statement is an equation or an identity, providing a mathematical argument: (3x + 10) / (x2 - 4) = 4 / (x - 2) - 1 / (x + 2)
- 1
The statement is an identity if the Right-Hand Side (RHS) can be shown to be equivalent to the Left-Hand Side (LHS).
Let's simplify the RHS.
- 2
The RHS is 4/(x-2) - 1/(x+2).
To combine these fractions, we need a common denominator.
- 3
The common denominator is (x-2)(x+2), which is a difference of two squares, (x2 - 4).
- 4
Rewrite the RHS with the common denominator:
[4(x+2) - 1(x-2)] / [(x-2)(x+2)].
- 5
Expand the brackets in the numerator:
(4x + 8 - x + 2) / (x2 - 4).
- 6
Collect like terms in the numerator:
(3x + 10) / (x2 - 4).
- 7
This simplified form of the RHS is identical to the LHS.
- 8
Since the RHS is algebraically equivalent to the LHS for all valid values of x (where x is not 2 or -2), the statement is an identity.
Answer: The statement is an identity.
Common mistakes
- ×Sign errors when distributing a negative. For instance, in `-(x-2)`, forgetting to distribute the negative to the `-2` to get `-x + 2`.
- ×Treating an identity proof like solving an equation. You must not move terms from one side to the other; instead, simplify one side until it matches the other.
- ×Errors in algebraic fraction manipulation, such as finding an incorrect common denominator or only multiplying the denominator and not the numerator.
No-calculator tips
- ✓To quickly check if a statement is NOT an identity, substitute a simple number like x=0 or x=1 into both sides. If the results are different, it must be an equation.
- ✓When proving an identity, always choose to simplify the more complicated-looking side. It's far easier to simplify an expression than to make a simple one more complex in a specific way.
- ✓For expressions involving `(x-a)(x+a)`, immediately recognise this as the difference of two squares, `x2 - a2`, to save time on expansion.