Lesson 3.2: Slope of a Line

New Concept

This lesson introduces slope ($m$), the measure of a line's steepness, calculated as $m = \frac{\text{rise}}{\text{run}}$. You'll learn to graph lines, interpret real-world applications, and identify parallel or perpendicular relationships using this powerful concept.

What’s next

Next, you'll master this through worked examples, interactive practice cards, and short videos, building your skills for graphing and analyzing lines.

Property

The slope of a line is $m = \frac{\text{rise}}{\text{run}}$. The rise measures the vertical change and the run measures the horizontal change. To find the slope from a graph, locate two integer-coordinate points, sketch a right triangle between them, and count the rise (vertical change) and run (horizontal change) to form the ratio.

Examples

• A line passes through $(1, 2)$ and $(4, 8)$. The rise is $8-2=6$ and the run is $4-1=3$. The slope $m = \frac{6}{3} = 2$.
• From a graph, you count that a line goes down 4 units (rise = -4) for every 2 units it moves to the right (run = 2). The slope is $m = \frac{-4}{2} = -2$.
• A roof rises 5 feet for every 15 feet of horizontal distance. The slope (pitch) of the roof is $m = \frac{5}{15} = \frac{1}{3}$.

Explanation

Slope measures a line's steepness. It's the ratio of how much the line goes up or down (rise) for every unit it moves left or right (run). A positive slope goes up from left to right, while a negative slope goes down.

Property

The slope of the line between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

Examples

• Find the slope between $(2, 3)$ and $(7, 9)$. Using the formula: $m = \frac{9 - 3}{7 - 2} = \frac{6}{5}$.
• Find the slope between $(-1, 5)$ and $(3, -3)$. Using the formula: $m = \frac{-3 - 5}{3 - (-1)} = \frac{-8}{4} = -2$.
• Find the slope between $(-4, -2)$ and $(-6, 8)$. Using the formula: $m = \frac{8 - (-2)}{-6 - (-4)} = \frac{10}{-2} = -5$.

Explanation

The slope formula is a way to calculate rise over run without a graph. It finds the vertical distance between points $(y_2 - y_1)$ and divides it by the horizontal distance $(x_2 - x_1)$ to find the steepness.

Property

• The slope of a horizontal line, $y=b$, is 0.
• The slope of a vertical line, $x=a$, is undefined.

Examples

• The slope of the horizontal line $y = 6$ is $m=0$.
• The slope of the vertical line $x = -2$ is undefined.
• The slope between points $(5, -1)$ and $(-3, -1)$ is $m = \frac{-1 - (-1)}{-3 - 5} = \frac{0}{-8} = 0$. The line is horizontal.

Explanation

A horizontal line is perfectly flat, so its 'rise' is always 0, making the slope 0. A vertical line is perfectly steep, so its 'run' is 0. Since we can't divide by zero, the slope is called undefined.

Property

To graph a line given a point and the slope:

1. Plot the given point.
2. Use the slope formula $m = \frac{\text{rise}}{\text{run}}$ to identify the rise and the run.
3. Starting at the given point, count out the rise and run to mark the second point.
4. Connect the points with a line.

Examples

• Graph a line through $(2, 1)$ with slope $m = \frac{2}{3}$. Start at $(2, 1)$, rise 2 units to $(2, 3)$, then run 3 units to $(5, 3)$. Connect $(2, 1)$ and $(5, 3)$.
• Graph a line through $(-1, 4)$ with slope $m = -3 = \frac{-3}{1}$. Start at $(-1, 4)$, fall 3 units (rise of -3) to $(-1, 1)$, then run 1 unit to $(0, 1)$. Connect the points.
• Graph a line through $(0, -2)$ with slope $m = \frac{1}{2}$. Start at $(0,-2)$, rise 1 unit to $(0, -1)$, then run 2 units to $(2, -1)$. Connect the points.

Explanation

Think of this as giving directions. Start at your known location (the point). The slope tells you where to go next: up or down for the rise, then right for the run. Mark your new spot and draw the line between them.

Property

The slope-intercept form of an equation of a line with slope $m$ and y-intercept, $(0, b)$ is

Examples

• The equation $y = 3x + 5$ has a slope $m=3$ and a y-intercept at $(0, 5)$.
• For $2x + y = 7$, first solve for y: $y = -2x + 7$. The slope is $m=-2$ and the y-intercept is $(0, 7)$.
• For $8x - 4y = 12$, solve for y: $-4y = -8x + 12$, so $y = 2x - 3$. The slope is $m=2$ and the y-intercept is $(0, -3)$.

Explanation

This form is a shortcut for graphing. By solving an equation for $y$, you can immediately identify the slope ($m$) and the starting point on the y-axis ($b$). This makes graphing quick and easy without plotting extra points.

Property

Strategy for Choosing the Most Convenient Method to Graph a Line:

• One variable: If the equation is $x=a$ or $y=b$, it is a vertical or horizontal line.
• $y$ is isolated: If the equation is in the form $y=mx+b$, use the slope and y-intercept.
• $Ax+By=C$ form: If $x$ and $y$ are on the same side, find the x- and y-intercepts.

Examples

• For $x = 5$, the most convenient method is to draw a vertical line passing through 5 on the x-axis.
• For $y = \frac{2}{3}x - 1$, the best method is to use the slope-intercept form. Start at $(0, -1)$, then rise 2 and run 3.
• For $3x + 4y = 12$, the easiest method is to find the intercepts. The x-intercept is $(4, 0)$ and the y-intercept is $(0, 3)$.

Explanation

Choosing the right tool makes the job easier. By looking at the equation's structure, you can pick the fastest graphing method instead of always plotting points, which can be slow. This strategy helps you work smarter, not harder.

Property

Parallel Lines are lines in the same plane that do not intersect. They have the same slope and different y-intercepts. If $m_1$ and $m_2$ are the slopes of two parallel lines, then $m_1 = m_2$.

Perpendicular Lines are lines in the same plane that form a right angle. Their slopes are negative reciprocals of each other, meaning $m_1 = -\frac{1}{m_2}$, and their product is $-1$, so $m_1 \cdot m_2 = -1$.

Examples

• The lines $y = 2x + 3$ and $y = 2x - 5$ are parallel because both have a slope of $m=2$.
• The lines $y = \frac{3}{4}x + 1$ and $y = -\frac{4}{3}x + 9$ are perpendicular because their slopes, $\frac{3}{4}$ and $-\frac{4}{3}$, are negative reciprocals.
• The lines $y = 5x - 1$ and $y = -5x + 2$ are neither parallel nor perpendicular because their slopes are not equal and not negative reciprocals.