Linear Equations and Inequalities: O-Level / IGCSE Maths (0580)

This topic brings together the algebra of straight-line relationships: solving linear equations in one unknown, solving simultaneous linear equations in two unknowns, and solving and representing linear inequalities. A linear equation such as $3x - 7 = 11$ contains an unknown only to the first power, and the goal is to isolate that unknown by applying the same operation to both sides until it stands alone, for example adding $7$ then dividing by $3$ to obtain $x = 6$. The same balancing idea extends to equations with brackets, fractions, or the unknown appearing on both sides.

When two unknowns are involved, a single equation is not enough to pin down their values, so we work with a pair of equations at once. The two standard techniques are elimination, where the equations are added or subtracted (after scaling if needed) to remove one variable, and substitution, where one equation is rearranged to make one variable the subject and then put into the other. Both methods reduce the problem to a single linear equation, after which the second value follows; the solution $(x, y)$ is the point where the two lines meet. Inequalities are handled much like equations, with one key rule: multiplying or dividing by a negative number reverses the inequality sign, as in $-2x > 6 \Rightarrow x < -3$. Solutions are often shown on a number line, using an open circle for strict inequalities ($<$, $>$) and a filled circle for inclusive ones ($\le$, $\ge$).

Mastering these methods builds directly on rearranging formulae and underpins later work on graphs, regions, and problem-solving across the syllabus. The worked examples below are original, written from the syllabus objective to illustrate each method with full, step-by-step solutions.