Equations, identities and algebraic argument methods
Algebra • Notation, vocabulary and manipulation
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Key concepts
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Definition: equation versus identity
An equation states that two expressions are equal for specific values of the variable(s). Solutions appear by finding variable values that satisfy the equality. An identity states that two expressions are equal for every value in their common domain. Verification of an identity requires transformation or simplification that holds generally, not just for particular values.
Techniques for showing equivalence
Algebraic equivalence arises from valid operations: expanding, factorising, collecting like terms, and simplifying fractions. Each operation preserves equality when applied to both sides of an equation or to an expression being transformed. Rational manipulation requires attention to domains because division by an expression that can be zero changes the set of valid values.
Proof by algebraic manipulation
A direct algebraic proof rewrites one side into the other using reversible steps. Each step states the operation and the condition that preserves equality, which produces a chain of logically connected expressions. The conclusion follows when the rewritten form matches the target expression, and the domain conditions confirm general validity.
Using substitution and counterexample
Substitution tests whether an equality holds for chosen values. One valid example cannot prove an identity because a single case does not guarantee generality. A counterexample disproves an identity or a proposed equation claim by demonstrating a specific value that violates the equality.
Domain considerations and limiting factors
Algebraic statements require explicit attention to domain restrictions such as denominators not equalling zero or even/odd root domains. Ignoring domain restrictions causes invalid arguments: for example, multiplying both sides by an expression that may be zero can introduce extraneous solutions or remove solutions. Statements about identities must specify the domain where the equivalence holds.
Constructing algebraic arguments
A robust algebraic argument lists starting assumptions, performs justified algebraic steps, and states domain conditions and conclusions. Cause leads to effect: each valid algebraic operation causes a new form that preserves equality, and the accumulated steps produce the final result. Clear justification of each step converts symbolic work into a mathematical proof.
Key notes
Important points to keep in mind