Unveiling the Slope: The Line Connecting (3, 2) and (10, 2)
What is the slope of the line that passes through (3, 2) and (10, 2)? This question delves into the fundamental concept of slope in linear equations, a crucial aspect of understanding how lines behave on a coordinate plane.
Editor's Note: This article provides a clear and concise explanation of calculating the slope of a line given two points, using a stepbystep approach and highlighting the importance of this fundamental concept in mathematics.
The slope of a line is a measure of its steepness, indicating how much the line rises or falls for every unit change in the horizontal direction. Understanding slope is essential in various fields like engineering, physics, and economics, where it helps model relationships between variables.
Analysis: To determine the slope of a line, we employ the slope formula, which utilizes the coordinates of two points on the line:
Slope (m) = (y2  y1) / (x2  x1)
Where:
 (x1, y1) and (x2, y2) represent the coordinates of the two points.
Key Takeaways of Slope
Key Takeaway  Description 

Positive Slope: Line rises from left to right.  
Negative Slope: Line falls from left to right.  
Zero Slope: Line is horizontal.  
Undefined Slope: Line is vertical. 
Let's apply this formula to our given points (3, 2) and (10, 2):

Label the points:
 (x1, y1) = (3, 2)
 (x2, y2) = (10, 2)

Substitute the values into the slope formula:
 m = (2  2) / (10  (3))

Simplify the equation:
 m = (4) / (13)
 m = 4/13
Therefore, the slope of the line passing through (3, 2) and (10, 2) is 4/13.
Understanding the Result:
 Negative Slope: The negative slope indicates that the line falls from left to right. This aligns with the visual representation of the points on a coordinate plane, where the ycoordinate decreases as the xcoordinate increases.
 Magnitude: The magnitude of the slope, 4/13, reflects the rate of change. For every 13 units moved horizontally to the right, the line falls 4 units vertically.
Exploring Further:
The Equation of the Line
Knowing the slope allows us to determine the equation of the line using the pointslope form:
y  y1 = m(x  x1)
Substituting the slope (4/13) and one of the points (let's use (3, 2)), we get:
 y  2 = (4/13)(x  (3))
 y  2 = (4/13)(x + 3)
This equation represents the line passing through the points (3, 2) and (10, 2).
Applications of Slope
The concept of slope has broad applications in various fields:
 Physics: Understanding the slope of a velocitytime graph helps determine acceleration.
 Engineering: Engineers use slope to analyze the stability of structures and design roads and bridges.
 Economics: Slope is used to analyze supply and demand curves, indicating changes in price and quantity.
In Conclusion: Calculating the slope of a line using two given points is a fundamental skill in mathematics, with implications across various disciplines. Understanding the slope provides valuable insights into the behavior of lines, enabling us to analyze and model realworld phenomena.