Concepts: forms of the equation of a line

3.1 Horizontal, vertical, point-slope and two-point forms

A horizontal line at distance a from the x-axis is written y equal to a, and a vertical line at distance b from the y-axis is written x equal to b. These are the only lines whose equations miss one variable, and recognising them saves work: a horizontal line has slope 0 and a vertical line has no slope.

The point-slope form is the workhorse. With a point (x1, y1) and slope m, the line is y minus y1 equal to m times (x minus x1). The two-point form is the point-slope form with the slope computed from the two given points. Everything else in the chapter reduces to these two: a parallel or perpendicular condition supplies the slope, and a distance condition supplies an equation to solve.

3.2 Slope-intercept and intercept forms

The slope-intercept form is y equal to mx plus c. Here m is the slope and c is the y-intercept, the y-value where the line crosses the y-axis. Reading a line in this form is immediate: the coefficient of x is the slope and the constant is the intercept. Converting a general equation to this form means solving for y, which shows the slope as minus A by B.

The intercept form is x by a plus y by b equal to 1, where a is the x-intercept and b is the y-intercept. It is the quickest form when the two intercepts are given, and it shows both intercepts without solving. Watch the signs: a negative intercept is written with its sign, so a line cutting the negative y-axis at 3 has b equal to minus 3, not 3.

The intercept form also handles questions about a line cutting equal intercepts on the axes. If a equals b, the line is x plus y equal to a, and its slope is minus 1; if the intercepts are equal in magnitude but opposite in sign, the slope is plus 1. This is where many objective questions come from.

A useful conversion check is to substitute both intercepts back into the equation: a line with intercepts a and b must give y equal to zero at x equal to a and x equal to zero at y equal to b. If either substitution fails, the equation has been mis-simplified, and the error is almost always in the constant term.

3.3 The general equation and special conditions

Every line can be written as Ax plus By plus C equal to 0 with A and B not both zero. From this form, the slope is minus A by B, the x-intercept is minus C by A and the y-intercept is minus C by B. The line is horizontal if A is zero and vertical if B is zero, because those are the cases where the corresponding intercept formula breaks down.

A point lies on a line exactly when its coordinates satisfy the equation. This simple test appears in parameter questions: substitute the point, solve for the parameter. Two lines are the same line when their coefficient triples are proportional, and they are parallel without being identical when only the A and B ratios match.

When a question gives a family of lines, such as lines through the intersection of two given lines, write the family as a combination of the two equations. The correct choice of parameter then satisfies whatever extra condition the question adds.

When a line passes through a fixed point and also satisfies a second condition, substituting the point first is usually cheaper than writing the whole equation. It turns two unknowns into one and often makes the family of lines obvious, which is exactly what a locus question needs.

3.4 Exercise 9.2 practice

Exercise 9.2 asks for equations of lines through given points with given slopes, lines through two points, lines parallel or perpendicular to given lines, and lines with given intercepts. Question 4 asks for the line through (1, 2) with slope 3, which is y equal to 3x minus 1. Question 5 asks for the line through (3, 4) with slope 1 by 2, which is x minus 2y plus 5 equal to 0.

The intercept questions are quick once the form is right. A line cutting x-intercept 3 and y-intercept 2 is x by 3 plus y by 2 equal to 1, which becomes 2x plus 3y equal to 6. A line through (2, 3) cutting equal intercepts on the axes is x plus y equal to 5, because equal intercepts force a slope of minus 1.

The perpendicular questions use the negative reciprocal slope. A line perpendicular to 3x minus 4y plus 7 equal to 0 has slope minus 4 by 3, since the original slope is 3 by 4. Substituting the given point in the point-slope form then gives the answer, and simplifying to general form avoids the fractions.