Maths: Linear Equations and Intersections for IB Maths AA SL

Straight-line geometry is the bridge between algebra and shape, letting us describe motion, trends, and spatial relationships with a single equation. In Maths AA SL, mastering linear relationships means understanding how a gradient and a point define a unique line, and how two lines can relate to each other—either crossing at a single point or never meeting at all. At the heart of this topic is the point-gradient form, y − y₁ = m(x − x₁), which builds a line from its slope and a known coordinate. Rearranging into y = mx + c reveals the y-intercept, while setting y = 0 finds the x-intercept—where the line crosses the horizontal axis. The real power emerges when comparing two lines: equal gradients signal parallel lines that never intersect, while different gradients guarantee a unique intersection point. This interconnected logic—from one line’s equation to its intercepts, then to relationships between lines—forms a toolkit for solving geometric problems algebraically, a skill that reappears across calculus, kinematics, and optimisation.