Interpret the slope of a linear relation

Slope represents the constant rate of change in a linear relation: the ratio of the change in the dependent variable to the corresponding change in the independent variable, $m=\Delta y/\Delta x$ . From graphs, tables, equations, and contexts, this understanding includes interpreting positive, negative, and zero slopes, attaching appropriate units, and distinguishing slope from a point’s coordinates or the $y$ -intercept. It supports proportional reasoning and linear modeling; instantaneous rates of change and calculus-based interpretations are not included.

Detailed Explanation: Interpret the slope of a linear relation

Slope tells how much the dependent variable changes for each 1-unit change in the independent variable:

m=\frac{\Delta y}{\Delta x}=\frac{\text{change in }y}{\text{change in }x}.

Example

A taxi’s total fare depends linearly on the distance traveled.

Distance, $x$ (km)	Fare, $y$ (dollars)
$2$	$7$
$5$	$13$

Step 1: Identify the variables.

Independent variable: distance, $x$
Dependent variable: fare, $y$

Step 2: Find the changes.

From $2$ km to $5$ km:

\Delta x=5-2=3\text{ km}

From 7 $to$ $13$:

\Delta y=13-7=\$6

Step 3: Calculate the slope.

m=\frac{\Delta y}{\Delta x}=\frac{\$6}{3\text{ km}}=\$2\text{ per km}

Step 4: Interpret the slope.

The slope is 2\text{ per km}} $. This means the fare increases by$ $2$ for every additional kilometre traveled.

The slope is not the point $(2,7)$ . That point means the fare is 7 $when the distance is$ 2$ km. The slope describes the constant rate of change between points.

A positive slope means $y$ increases as $x$ increases.
A negative slope means $y$ decreases as $x$ increases.
A zero slope means $y$ stays constant as $x$ increases.

Always include units when interpreting a slope: use the units of $y$ divided by the units of $x$ .

Learn by doing: Interpret the slope of a linear relation

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Slope - Find Equivalent - Graph to Decimal Slope

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