Linear Relations

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

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

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.

Recognizing a linear table

For equally spaced $x$-values, a linear relation has constant first differences in $y$. If input spacing is unequal, calculate $\Delta y/\Delta x$ between pairs instead.

A constant ratio $y/x$ signals direct proportion, which is the special linear case $b=0$.

Worked example: context to equation

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

Graphing a line

Plot the intercept $(0,b)$, then use slope as rise/run. For $y=2x-3$, start at $(0,-3)$ and move right $1$, up $2$ repeatedly.

Two distinct points determine a line. Label scales consistently and draw the line only across the meaningful contextual domain when appropriate.

Equation from two points

Find slope, then use point-slope form:

For points $(2,5)$ and $(6,13)$,

so $y-5=2(x-2)$ and $y=2x+1$.

Other equation forms

Standard form is

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.

Intersections

The intersection of two lines is an ordered pair satisfying both equations. Find it by graphing, substitution, or elimination.

Different slopes give one intersection, equal slopes/different intercepts give none, and equivalent equations give infinitely many.

Domain and range

An unrestricted line extends infinitely, but real contexts may restrict inputs. Time may be nonnegative; item counts may be whole numbers; capacity may create an upper bound.

Graph discrete points when only separate counts are allowed.

Interpolation and extrapolation

Interpolation predicts inside known data; extrapolation predicts beyond it and assumes the same rate continues. A linear model may be useful over one interval and unrealistic far outside it.

State the model's evidence and limitations.

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?

Related topics

• Slope
• Slope-Intercept Form
• Proportional Reasoning
• Systems of Linear Equations