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Finding Slope From Two Points: Definition and Examples | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">Finding Slope From Two PointsFinding Slope From Two Points: Definition and ExamplesTable of ContentsFinding Slope of Line from Two Points Definition of Finding Slope from Two Points

Finding slope from two points refers to calculating the slope of a line when the coordinates of two points on the line are given. The slope is a measurement of the change in vertical distance over the change in horizontal distance, often called the rise-over-run ratio. It tells us how steep a straight line is and can be positive (upward slope), negative (downward slope), zero (horizontal line), or undefined (vertical line).

The formula for finding slope from two points (x1,y1)(x_1, y_1)(x1​,y1​) and (x2,y2)(x_2, y_2)(x2​,y2​) is derived from the basic principle of slope as tangent of the angle formed with the positive x-axis. By drawing a right triangle between the two points and using trigonometric principles, we can establish that the slope equals m=y2−y1x2−x1m = \frac{y_2-y_1}{x_2-x_1}m=x2​−x1​y2​−y1​​, where the numerator represents the vertical change (rise) and the denominator represents the horizontal change (run).

Examples of Finding Slope from Two Points Example 1: Finding Slope Between Two Coordinate Points Problem:

Calculate the slope of a line passing through the points P(  –1,  3)P(\;–1,\; 3)P(–1,3) and Q(2,  15)Q(2,\; 15)Q(2,15).

Step-by-step solution:

Step 1, Identify the coordinates of both points. For point P, (x1,  y1)=(  –1,  3)(x_1,\; y_1) = (\;–1,\; 3)(x1​,y1​)=(–1,3) and for point Q, (x2,  y2)=(2,  15)(x_2,\; y_2) = (2,\; 15)(x2​,y2​)=(2,15).

Step 2, Set up the slope formula. We will use m=y2−y1x2−x1m = \frac{y_2-y_1}{x_2-x_1}m=x2​−x1​y2​−y1​​, where we plug in our coordinates.

Step 3, Calculate the difference in y-coordinates (rise). y2−y1=15−3=12y_2 - y_1 = 15 - 3 = 12y2​−y1​=15−3=12

Step 4, Calculate the difference in x-coordinates (run). x2−x1=2−(−1)=2+1=3x_2 - x_1 = 2 - (-1) = 2 + 1 = 3x2​−x1​=2−(−1)=2+1=3

Step 5, Divide rise by run to find the slope. m=123=4m = \frac{12}{3} = 4m=312​=4

Hence, the slope of the line PQ is 4.

Example 2: Finding Slope with Different Units Problem:

What is the slope of a line if rise =100= 100=100 inches and run =25= 25=25 feet?

Step-by-step solution:

Step 1, Notice that rise and run are in different units. We need to convert them to the same unit before finding the slope.

Step 2, Convert run from feet to inches. Since 1 foot =12= 12=12 inches, we have: 252525 feet =25×12=300= 25 \times 12 = 300=25×12=300 inches

Step 3, Calculate the slope using the rise-over-run formula. Slope =RiseRun=100300=13= \frac{\text{Rise}}{\text{Run}} = \frac{100}{300} = \frac{1}{3}=RunRise​=300100​=31​

Hence, the slope of the line is 13\frac{1}{3}31​.

Example 3: Finding an Unknown Value Using Slope Problem:

If the slope of the line joining two points (  –1,  2k)(\;–1,\; 2k)(–1,2k) and (  –2,  k+2)(\;–2,\; k+2)(–2,k+2) is –2–2–2, then find the value of k.

Step-by-step solution:

Step 1, Identify what we know. The slope of the line is −2-2−2, and the line passes through the points (  –1,  2k)(\;–1,\; 2k)(–1,2k) and (  –2,  k+2)(\;–2,\; k + 2)(–2,k+2).

Step 2, Label the coordinates. Let (x1,  y1)=(  –1,  2k)(x_1,\; y_1) = (\;–1,\; 2k)(x1​,y1​)=(–1,2k) and (x2,  y2)=(  –2,  k+2)(x_2,\; y_2) = (\;–2,\; k + 2)(x2​,y2​)=(–2,k+2).

Step 3, Write the slope using the formula. Slope=y2−y1x2−x1\text{Slope} = \frac{y_2-y_1}{x_2-x_1}Slope=x2​−x1​y2​−y1​​

Step 4, Substitute the known values and set up an equation. −2=(k+2)−2k(−2)−(−1)-2 = \frac{(k + 2) - 2k}{(-2) - (-1)}−2=(−2)−(−1)(k+2)−2k​

Step 5, Simplify the equation. −2=k+2−2k−2−(−1)=−k+2−1=k−21-2 = \frac{k + 2 - 2k}{-2 - (-1)} = \frac{-k + 2}{-1} = \frac{k - 2}{1}−2=−2−(−1)k+2−2k​=−1−k+2​=1k−2​

Step 6, Solve for k.

−2=k−2-2 = k - 2−2=k−2 k−2=−2k - 2 = -2k−2=−2 k=0k = 0k=0

Hence, the value of k is 0.

Comments(2)TTableTennisFanXavierNovember 6, 2025I've been struggling to teach slope from two points. This page's clear def and examples made it click for my students. Thanks!

DDadOf3BoysNovember 4, 2025I've been struggling to explain slope to my students. This page's def and examples made it super easy! Thanks for the great resource.

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