Linear Relations

A linear relation has a constant rate of change and can be represented by a table, graph, equation, or context.

Slope compares vertical change with horizontal change. A line contains two marked points joined by a right-angled rise-and-run triangle. The vertical and horizontal legs make the signed ratio behind slope visible.

Linear relations change by equal output amounts over equal input intervals, producing straight-line graphs.

Constant rate of change

For points $(x_1,y_1)$ and $(x_2,y_2)$ on a nonvertical line, slope is

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

It measures output change per one input unit. A positive slope rises left to right, a negative slope falls, zero is horizontal, and a vertical line has undefined slope.

Slope-intercept form

A nonvertical linear relation can be written

$$ y=mx+b, $$

where $m$ is slope and $b$ is the $y$-intercept. The intercept is the output when $x=0$.

Units matter: slope uses output units per input unit; intercept uses output units.

Worked example: context to equation

This relation is linear but not proportional because it has a nonzero base fee.

Other equation forms

Standard form is

$$ Ax+By=C. $$

Vertical lines use $x=c$ and cannot be written as $y=mx+b$. Horizontal lines use $y=c$ and have slope zero.

Choose a form that reveals the information needed.

Common mistakes

Computing run over rise. Slope is vertical change divided by horizontal change.

Reversing subtraction in only one part. Keep point order consistent.

Calling every line proportional. Proportional graphs pass through the origin.

Using $b$ as an $x$-intercept. It is the output at $x=0$.

Ignoring contextual domain. Mathematical continuation may be meaningless.

Quick self-check

Is rate of change constant?
What do slope and intercept mean with units?
Do table, graph, equation, and context agree?
Is the line proportional or only linear?
Are vertical/horizontal special cases handled correctly?
Does the domain reflect the real situation?