What Is The Slope Of (6,5) And (3,7.5) ??
Understanding the concept of slope is fundamental in coordinate geometry, as it measures the steepness or inclination of a line connecting two points on a Cartesian plane. When given two points, such as (6, 5) and (3, 7.5), calculating the slope provides insight into how the line behaves—whether it rises, falls, or remains horizontal. This article explores the definition of slope, how to compute it, and applies this understanding to the specific points (6, 5) and (3, 7.5).
---
What Is the Slope in Coordinate Geometry?
Definition of Slope
The slope of a line, often denoted by the letter 'm', quantifies the rate at which the y-coordinate changes with respect to the x-coordinate along that line. Essentially, it tells us how steep the line is and whether it ascends or descends as we move from left to right.Mathematically, the slope between two points \((x1, y1)\) and \((x2, y2)\) is defined as:
\[
m = \frac{y2 - y1}{x2 - x1}
\]
This ratio indicates the change in the vertical direction (rise) over the change in the horizontal direction (run).
Significance of the Slope
- A positive slope indicates the line rises from left to right.
- A negative slope signifies the line falls from left to right.
- A zero slope corresponds to a horizontal line.
- An undefined slope occurs when the line is vertical, i.e., when \(x2 - x1 = 0\).
---
Calculating the Slope Between (6, 5) and (3, 7.5)
Step-by-Step Calculation
To find the slope between these two points, we identify the coordinates:- Point 1: \((x1, y1) = (6, 5)\)
- Point 2: \((x2, y2) = (3, 7.5)\)
\[
m = \frac{7.5 - 5}{3 - 6}
\]
Calculate numerator (change in y):
\[
7.5 - 5 = 2.5
\]
Calculate denominator (change in x):
\[
3 - 6 = -3
\]
Now, compute the slope:
\[
m = \frac{2.5}{-3} = -\frac{2.5}{3}
\]
Expressed as a decimal:
\[
m \approx -0.8333
\]
Or as a simplified fraction:
\[
m = -\frac{5}{6}
\]
Therefore, the slope of the line passing through the points (6, 5) and (3, 7.5) is \(-\frac{5}{6}\) or approximately \(-0.8333\).
Interpretation of the Result
- The negative sign indicates the line descends as it moves from left to right.
- The magnitude \( \frac{5}{6} \) suggests that for every 6 units moved horizontally, the line drops approximately 5 units vertically.
Understanding the Significance of the Slope Value
What Does a Slope of \(-\frac{5}{6}\) Indicate?
This specific slope value conveys several key points:- The line has a gentle downward incline.
- The line is decreasing; as the x-value increases, the y-value decreases at a rate of about 0.8333 units per unit increase in x.
- The line’s steepness is less than 1, indicating a relatively moderate slope.
How to Visualize This Slope
Imagine plotting the points:- Starting at \((6, 5)\), if you move 6 units to the left (to \(x=0\)), the y-value would increase by approximately 5 units (since the slope is negative, moving left would increase y).
- Conversely, moving from \((6, 5)\) to \((3, 7.5)\), the y-coordinate increases from 5 to 7.5 while x decreases from 6 to 3, reflecting the negative slope.
Additional Concepts Related to Slope
Equation of the Line Through the Two Points
Once the slope is known, the equation of the line can be written in point-slope form:\[
y - y1 = m(x - x1)
\]
Using point \((6, 5)\):
\[
y - 5 = -\frac{5}{6}(x - 6)
\]
Simplify and rearrange to slope-intercept form \(y = mx + b\):
\[
y - 5 = -\frac{5}{6}x + 5
\]
Adding 5 to both sides:
\[
y = -\frac{5}{6}x + 10
\]
This equation describes the line passing through the two points.
Verifying the Equation with the Second Point
Substitute \(x=3\):\[
y = -\frac{5}{6} \times 3 + 10 = -\frac{15}{6} + 10 = -\frac{5}{2} + 10 = -2.5 + 10 = 7.5
\]
This matches the y-coordinate of the second point, confirming the correctness of the line’s equation.
---
Applications of Slope in Real-World Contexts
Engineering and Construction
- Calculating the slope of roads and ramps.
- Designing inclined surfaces and ensuring safety standards.
Economics and Business
- Analyzing the rate of change of costs or revenues with respect to production levels.
Science and Nature
- Describing the rate of change in physical phenomena, such as velocity or growth rates.
Data Analysis
- Fitting lines to data points for predictive modeling.
Summary and Key Takeaways
- The slope between two points measures how steep a line is and whether it rises or falls.
- For points (6, 5) and (3, 7.5), the slope is \(-\frac{5}{6}\), indicating a moderate downward incline.
- The slope can be used to derive the line’s equation, which aids in graphing and analysis.
- Understanding slope has practical applications across various fields, from engineering to economics.
