Equations with Brackets & Fractions

Expansion Method: To remove brackets, multiply the term outside the bracket by every term inside. For example, $k(ax + b)$ becomes $kax + kb$.

Handling Negatives: If the multiplier is negative, the signs of all terms inside the bracket must be reversed during expansion. A common error is only changing the sign of the first term.

Division Shortcut: If the entire side of an equation is multiplied by a constant, you can sometimes divide both sides by that constant first. This is efficient if the constant is a factor of the other side's terms.

Moving the Unknown: When the variable $x$ appears in the denominator, such as $\frac{a}{x+b} = c$, multiply both sides by the entire denominator expression $(x+b)$.

Linear Transformation: This action moves the variable to the 'numerator level' on the opposite side, often creating a bracketed expression like $a = c(x+b)$ which can then be expanded.

Restriction Check: Always ensure the final solution does not make the original denominator zero, as division by zero is undefined.

The Substitution Test: Always verify your final answer by substituting it back into the original equation. If the left-hand side equals the right-hand side, the solution is correct.

Fractional Answers: Do not be alarmed if the result is a fraction or decimal. Unless specified, improper fractions like $\frac{7}{3}$ are usually preferred over rounded decimals.

Show All Steps: Examiners look for the 'clearing' step. Clearly state what you are multiplying by to avoid losing marks for arithmetic slips.