Coordinate Geometry (Advanced)

Slope measures how steep a line is. A straight line can be written in general, slope-intercept or intercept form, depending on what is known.

Key ideas

Reflect in the x-axis: y changes sign; in the y-axis: x changes sign; in the origin: both change sign.
The distance of (a, b) from the x-axis is |b| and from the y-axis is |a|.
Positive slope rises from left to right; negative slope falls; a horizontal line has slope 0; a vertical line has no slope.
Three points are on one line exactly when the slopes between them are equal.
Find intercepts by putting y = 0 (x-intercept) or x = 0 (y-intercept).
The signs of the two intercepts tell you which one quadrant a line never enters.

Slope through two points	m = (y₂ − y₁) ÷ (x₂ − x₁)
Parallel and perpendicular	m₁ = m₂; m₁m₂ = −1
General form Ax + By + C = 0	slope −A/B; x-intercept −C/A; y-intercept −C/B
Slope-intercept form	y = mx + c (c is the y-intercept)
Intercept form	x/a + y/b = 1 (cuts the axes at (a, 0) and (0, b))
Triangle with the axes	area = ½ × \|a\| × \|b\|

Subtracting the coordinates in different orders on the top and the bottom of the slope formula.
Taking the perpendicular slope as −m instead of −1/m.
Forgetting that a distance from an axis is never negative.

For the line −5x + 4y + 10 = 0, find the slope and both intercepts, and name the quadrant it never passes through.

Working: slope = −A/B; x-intercept = −C/A; y-intercept = −C/B; a line through (2, 0) and (0, −5/2) passes through the other three quadrants

Answer: slope 5/4, x-intercept 2, y-intercept −5/2; it misses quadrant II