Coordinate geometry: lines, slopes and intercepts

Slope is rise over run

The slope of the line through $(x_1, y_1)$ and $(x_2, y_2)$ is

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

It measures how much $y$ changes when $x$ increases by 1. Take both differences in the same order: if you start the top with the second point, start the bottom with the second point too. Mixing the order flips the sign.

Slope	The line
positive	rises from left to right
negative	falls from left to right
zero	horizontal, $y = c$
undefined	vertical, $x = c$

Three forms of a line

Slope-intercept form, $y = mx + b$: $m$ is the slope and $b$ is the $y$-intercept, where the line crosses the $y$-axis. This is the most useful form for reading a line at a glance.

Point-slope form, $y - y_1 = m(x - x_1)$: use it when you know one point $(x_1, y_1)$ and the slope. Then rearrange to $y = mx + b$.

Standard form, $Ax + By = C$: common in questions. To read its slope, solve for $y$: $y = -\frac{A}{B}x + \frac{C}{B}$, so the slope is $-\frac{A}{B}$. Forgetting the minus sign is the usual slip.

Intercepts

The $y$-intercept is where $x = 0$; the $x$-intercept is where $y = 0$. For $3x + 4y = 24$, setting $x = 0$ gives $y = 6$, and setting $y = 0$ gives $x = 8$. Standard form makes intercepts quick; slope-intercept form makes the slope quick. Choose the form that suits the question.

Is a point on the line?

A point lies on a line exactly when its coordinates satisfy the equation. To test $(4, 1)$ against $y = -\frac{1}{2}x + 3$, substitute $x = 4$: $y = -2 + 3 = 1$. It matches, so the point is on the line. This is often the fastest way through a question that lists points as options.

Parallel and perpendicular lines

• Parallel lines have equal slopes (and different intercepts).
• Perpendicular lines have slopes whose product is $-1$: each slope is the negative reciprocal of the other. A slope of $\frac{2}{3}$ pairs with $-\frac{3}{2}$; a slope of $-4$ pairs with $\frac{1}{4}$.

Both parts of "negative reciprocal" matter. Flipping the fraction without changing the sign, or changing the sign without flipping, gives a line that is not perpendicular. A horizontal line (slope 0) is perpendicular to a vertical line, which has no slope; this is the one case the product rule does not cover.

Midpoint and distance

The midpoint of $(x_1, y_1)$ and $(x_2, y_2)$ averages the coordinates:

$$\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

The distance between them comes from the Pythagorean theorem, with the horizontal and vertical gaps as the two legs:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Sketching the right triangle is often quicker than the formula, and it shows when the gaps form a familiar triple such as 6, 8, 10.

Regions on either side of a line

An inequality such as $y > 2x - 1$ describes the points above the line $y = 2x - 1$, and $y < 2x - 1$ the points below it. To check which side a region is on, test a convenient point such as $(0, 0)$: here $0 > -1$ is true, so the origin is in the region $y > 2x - 1$.

Draw it

Even a rough sketch on scratch paper catches sign errors. A line through $(-1, 4)$ and $(3, -4)$ obviously falls, so a positive slope is impossible; a perpendicular slope that has the same sign as the original slope is impossible too.