Formulae, subjects and substitution: Manipulate and substitute
Algebra • Notation, vocabulary and manipulation
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Key concepts
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Definition and role of a formula
A formula defines a mathematical relationship between symbols, where each symbol represents a number or measurement. A formula presents a rule that links quantities so that a change in one quantity causes predictable changes in others. A formula remains valid only within stated assumptions and domains. A formula that includes division requires nonzero denominators. A formula that includes square roots requires non-negative radicands when working with real numbers.
Subject of a formula
The subject of a formula is the variable isolated on one side of the equals sign, for example x in x = 2y + 3. Changing the subject produces an equivalent formula with a different isolated variable. A formula that makes a particular variable the subject clarifies how that variable depends on the others. Changing the subject requires operations that preserve equality. Each step must apply the same operation to both sides so that the relationship remains true for all allowed values.
Inverse operations and balancing
Addition and subtraction undo each other; multiplication and division undo each other; powers and roots undo each other. Rearrangement proceeds by applying inverse operations in the reverse order of operations. The effect of applying inverse operations is to isolate the desired variable. Every operation applied to one side must apply to the other side. The balance method preserves equality and prevents introduction of invalid solutions, provided domain restrictions are observed.
Rearranging linear formulae
Linear formulae contain variables to the first power. Rearrangement of linear formulae involves collecting terms containing the target variable, using addition or subtraction to move other terms, then using multiplication or division to isolate the subject. For example, from 3x + 2y = 12, subtraction of 2y from both sides gives 3x = 12 - 2y; division by 3 gives x = 4 - (2/3)y. Rearrangement of linear formulae produces a unique expression for the subject, provided division by zero does not occur.
Rearranging formulae with fractions, brackets and powers
Brackets require expansion or isolation before isolating the subject. Fractions require common denominators or multiplication by denominators to clear fractions. Powers require root operations applied carefully, and squaring both sides may introduce extraneous solutions that require checking. For example, from y = (2x + 1)/3, multiplication by 3 gives 3y = 2x + 1, then subtraction and division isolate x. Algebraic manipulation with fractions and powers must respect domain restrictions and avoid dividing by expressions that may equal zero.
Substitution and scientific formulae
Substitution replaces each variable in a formula with a given numerical value and then evaluates using arithmetic rules. Substitution in scientific formulae requires consistent units; mismatched units cause incorrect results. For example, density = mass ÷ volume requires mass and volume in compatible units before calculation. Substitution often requires rounding to an appropriate number of significant figures. The final answer must respect unit conventions and any given precision requirements.
Key notes
Important points to keep in mind