A Line Passes Through These Points: (0,6),(2,15) . What Is Its Slope? (provide One Decimal Place)
Understanding the concept of slope is fundamental in coordinate geometry, especially when analyzing lines passing through specific points. In this article, we will explore how to determine the slope of a line passing through the points (0,6) and (2,15). We will break down the steps involved in calculating the slope, explain the significance of the slope value, and provide practical examples to reinforce learning. Whether you’re a student preparing for exams or a math enthusiast, this guide will help you grasp the process clearly and accurately.
What Is the Slope of a Line?
Definition of Slope
The slope of a line measures how steep the line is. It describes the rate at which the y-coordinate (vertical change) changes in relation to the x-coordinate (horizontal change). Mathematically, the slope (often represented by the letter m) is given by:\[ m = \frac{\text{change in y}}{\text{change in x}} = \frac{Δy}{Δx} \]
This ratio indicates whether the line rises, falls, or remains horizontal as you move along it.
Importance of Slope in Coordinate Geometry
Knowing the slope helps in:- Predicting the behavior of the line (increasing, decreasing, or constant).
- Formulating the equation of the line.
- Understanding relationships between variables in real-world contexts.
How to Calculate the Slope Between Two Points
Given Points
Suppose you have two points on a coordinate plane:- Point 1: (x1, y1)
- Point 2: (x2, y2)
For our specific problem:
- Point 1: (0, 6)
- Point 2: (2, 15)
Applying the Slope Formula
The formula to find the slope (m) is:\[ m = \frac{y2 - y1}{x2 - x1} \]
Plugging in the given points:
\[
m = \frac{15 - 6}{2 - 0}
\]
Calculating numerator and denominator:
\[
m = \frac{9}{2}
\]
Which simplifies to:
\[
m = 4.5
\]
Therefore, the slope of the line passing through points (0,6) and (2,15) is 4.5.
Interpreting the Slope
What Does a Slope of 4.5 Mean?
A slope of 4.5 indicates that for every one unit increase in x (horizontal movement), y (vertical movement) increases by 4.5 units. This signifies a relatively steep line that ascends from left to right, reflecting a positive relationship between x and y.Visualizing the Line
- Starting at point (0,6), if you move 1 unit to the right along the x-axis, the y-value rises by approximately 4.5 units.
- Moving from (0,6) to (2,15), the y-value increases from 6 to 15, consistent with the slope calculation.
Formulating the Equation of the Line
Once the slope is known, the next step is often to write the line's equation in slope-intercept form:\[ y = mx + b \]
where:
- m is the slope,
- b is the y-intercept (the point where the line crosses the y-axis).
Finding the Y-Intercept (b)
Given the slope \( m = 4.5 \), and one point, say (0,6):
\[
6 = 4.5 \times 0 + b
\]
\[
b = 6
\]
Thus, the equation of the line is:
\[ y = 4.5x + 6 \]
This equation can be used to predict y-values for any x-value along the line.
Practical Applications of Slope Calculations
Real-World Contexts
Understanding and calculating slopes are essential in various fields:- Physics: Determining velocity when given displacement over time.
- Economics: Analyzing cost functions and rate of change.
- Engineering: Designing slopes and gradients for roads and ramps.
Educational Significance
Mastering slope calculations enhances problem-solving skills and lays the foundation for more advanced topics like linear equations, inequalities, and calculus.Summary and Key Takeaways
- The slope of a line passing through points (0,6) and (2,15) is 4.5.
- This slope indicates a steep, upward-sloping line.
- The formula for slope, \(\frac{Δy}{Δx}\), is straightforward to apply with two points.
- Using the slope and a point, you can derive the equation of the line in slope-intercept form.
- Calculating slopes is fundamental in understanding linear relationships across various disciplines.
