Determine the slope of a line

Slope is the constant rate of change of a nonvertical line, represented as rise over run and calculated from two points by $m=\frac{y_2-y_1}{x_2-x_1}$ , with the same value regardless of the points’ order when numerator and denominator are paired consistently. Its sign and magnitude describe direction and steepness; horizontal lines have slope zero, while vertical lines have undefined slope. The concept supports interpreting linear equations and proportional change, without extending to derivatives or other advanced rate-of-change ideas.

Detailed Explanation: Determine the slope of a line

Slope measures how steeply a line rises or falls. It is the rise over the run:

m=\frac{\text{change in }y}{\text{change in }x} =\frac{y_2-y_1}{x_2-x_1}

A positive slope means the line rises from left to right.
A negative slope means the line falls from left to right.
A slope of $0$ describes a horizontal line.
A vertical line has an undefined slope because its run is $0$ .

Example

Find the slope of the line through $(2,3)$ and $(6,11)$ .

Step 1: Identify the coordinates.

Let

(x_1,y_1)=(2,3) \quad\text{and}\quad (x_2,y_2)=(6,11).

Step 2: Substitute into the slope formula.

m=\frac{y_2-y_1}{x_2-x_1} =\frac{11-3}{6-2}

Step 3: Simplify.

m=\frac{8}{4}=2

The slope is

\boxed{2}.

This means that for every $1$ unit the line moves to the right, it rises $2$ units. Always keep the coordinates paired and subtract in the same order in the numerator and denominator. Using the points in the reverse order gives the same result:

\frac{3-11}{2-6}=\frac{-8}{-4}=2.

Learn by doing: Determine the slope of a line

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Line Segment (Graph) - Find Slope (Value)

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