3.1 Algebraic Formulae
An algebraic formula is an equation that relates a subject of the formula to other variables or constants through algebraic operations.
Subject of a Formula
- The subject of a formula is the variable that is expressed on one side of the equation (usually the left-hand side) on its own, with a coefficient of positive $1$.
- Example: In $y = 3x + 5$, $y$ is the subject of the formula.
Formulating an Algebraic Formula
To write an algebraic formula based on a given situation:
- Identify the dependent variable (the subject) and the independent variables.
- Represent each quantity using suitable variables (letters).
- Connect the variables using appropriate operational symbols ($+$, $-$, $\times$, $\div$, powers, roots).
Changing the Subject of a Formula
To change the subject of a formula to a different variable, use inverse operations while keeping the equation balanced:
- Addition $\leftrightarrow$ Subtraction: If $a + b = c$, then $a = c - b$.
- Multiplication $\leftrightarrow$ Division: If $ab = c$, then $a = \frac{c}{b}$.
- Squares $\leftrightarrow$ Square Roots: If $a^2 = b$, then $a = \sqrt{b}$.
- Cubes $\leftrightarrow$ Cube Roots: If $a^3 = b$, then $a = \sqrt[3]{b}$.
Determining the Value of a Variable
To find the value of a specific variable in a formula:
- Substitute the given numerical values for all known variables into the formula.
- Solve the resulting linear or quadratic equation to find the value of the target variable.
Solving Problems Involving Algebraic Formulae
Real-world problems involving areas, perimeters, speed, interest, and costs can be solved by rearranging formulae to isolate the required unknown and substituting the given measurements.