Learn the method, study an example, then explain your reasoning in an exam-style problem. Work on paper before opening the solution and compare each step with the marking guidance.
Straight-line equations, intercepts and intersections
Build a line equation from geometric information and use it in a system.
Notes
Before you start: find a gradient and solve two linear equations.
A nonvertical line with gradient m through \((x_0,y_0)\) satisfies \(y-y_0=m(x-x_0)\). Substitute the known point to check the constant. The form \(ax+by=c\) also represents vertical lines and avoids unnecessary fractions.
For an x-intercept, set y=0; for a y-intercept, set x=0. Do not exchange these substitutions. When two nonzero intercepts make a triangle with the origin, its area is half the product of their absolute values.
At an intersection both equations hold. Solve simultaneously, then substitute the paired coordinates into both original equations. Parallel distinct lines have no intersection; coincident equations describe infinitely many common points.
Recall later
A line of gradient −2 through \((1,3)\) is \(y=-2x+5\). Check the point before using the equation.
Common mistakes
Using y=mx+c without calculating c from the point.
Setting x=0 when finding an x-intercept.
Giving two separate coordinates without checking they form the same solution pair.
