U22 Change of Subject in Formulae: Essential Formulas

This topic focuses on rearranging a given formula to make a specific variable the subject. It is a fundamental algebraic skill required for solving problems in geometry, physics, and other applied areas of the DSE Mathematics curriculum.

1 Basic Principles and Steps

Core Concept

The subject of a formula is the variable that stands alone on one side of the equals sign. To change the subject, we perform inverse operations on both sides of the equation, following the reverse order of operations (BODMAS/PEMDAS). The goal is to isolate the new subject.

$$ \text{If } A = B + C, \text{ then } B = A - C $$
FormulaNew Subject

2 Common Formula Rearrangements

Linear Formulas

For formulas involving linear terms of the subject. Example: Make $x$ the subject of $y = mx + c$.

$$ y = mx + c \implies x = \frac{y - c}{m}, \quad m \neq 0 $$

Formulas Involving Squares and Square Roots

When the subject is squared or under a square root. Remember to consider the positive and negative roots where applicable. Example: Make $r$ the subject of $A = \pi r^{2}$.

$$ A = \pi r^{2} \implies r = \pm\sqrt{\frac{A}{\pi}}, \quad A \geq 0 $$

Formulas with Fractions

Often requires cross-multiplication. Example: Make $u$ the subject of the lens formula $\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$.

$$ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \implies u = \frac{fv}{v - f}, \quad v \neq f $$

3 Key Points and Common Mistakes

Important Notes

  • Perform the same operation to both sides of the equation to maintain balance.
  • The order of operations is crucial. Undo addition/subtraction before multiplication/division, and deal with powers/roots last.
  • State any restrictions on the variables (e.g., denominators cannot be zero, expressions under an even root must be non-negative).
  • When taking square roots, remember the $\pm$ symbol if both positive and negative roots are valid in context.
  • Check your final formula by substituting simple numbers to verify the rearrangement is correct.

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