What Is The Slope Of The Line That Goes Through The Points ( − 1 , 4 (-1,4 ( − 1 , 4 ] And ( 14 , − 2 (14,-2 ( 14 , − 2 ]?A. − 13 6 -\frac{13}{6} − 6 13 B. − 6 13 -\frac{6}{13} − 13 6 C. − 5 2 -\frac{5}{2} − 2 5 D. − 2 5 -\frac{2}{5} − 5 2
Introduction
When dealing with lines and their equations, understanding the concept of slope is crucial. The slope of a line is a measure of how steep it is and can be calculated using the coordinates of two points on the line. In this article, we will explore how to find the slope of a line given two points and apply this concept to a specific problem.
What is Slope?
The slope of a line is a numerical value that represents the rate of change of the line. It is calculated as the ratio of the vertical change (the change in the y-coordinate) to the horizontal change (the change in the x-coordinate) between two points on the line. The slope is often denoted by the letter 'm' and can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the coordinates of the two points on the line.
Calculating the Slope
To calculate the slope of the line that goes through the points (-1,4) and (14,-2), we can use the formula above. We will substitute the coordinates of the two points into the formula and perform the necessary calculations.
Step 1: Identify the Coordinates
The coordinates of the two points are (-1,4) and (14,-2). We will use these coordinates to calculate the slope.
Step 2: Calculate the Vertical Change
The vertical change is the difference between the y-coordinates of the two points. We will calculate this by subtracting the y-coordinate of the first point from the y-coordinate of the second point.
y2 - y1 = -2 - 4 = -6
Step 3: Calculate the Horizontal Change
The horizontal change is the difference between the x-coordinates of the two points. We will calculate this by subtracting the x-coordinate of the first point from the x-coordinate of the second point.
x2 - x1 = 14 - (-1) = 15
Step 4: Calculate the Slope
Now that we have calculated the vertical and horizontal changes, we can substitute these values into the formula to calculate the slope.
m = (y2 - y1) / (x2 - x1) m = (-6) / (15) m = -6/15 m = -2/5
Conclusion
The slope of the line that goes through the points (-1,4) and (14,-2) is -2/5. This means that for every unit of horizontal change, the line changes by -2 units of vertical change.
Answer
The correct answer is D. -2/5.
Discussion
The concept of slope is a fundamental concept in mathematics, particularly in geometry and algebra. Understanding how to calculate the slope of a line is crucial in solving problems involving lines and their equations. In this article, we have explored how to calculate the slope of a line given two points and applied this concept to a specific problem. We have also discussed the importance of understanding the concept of slope and its applications in mathematics.
Related Topics
- Calculating the slope of a line given two points
- Understanding the concept of slope
- Applications of slope in mathematics
- Geometry and algebra
Further Reading
References
Introduction
In our previous article, we explored the concept of slope and how to calculate it using the coordinates of two points on a line. In this article, we will answer some frequently asked questions about slope to help you better understand this concept.
Q: What is the slope of a horizontal line?
A: The slope of a horizontal line is 0. This is because the vertical change is 0, and the horizontal change is not defined.
Q: What is the slope of a vertical line?
A: The slope of a vertical line is undefined. This is because the horizontal change is 0, and the vertical change is not defined.
Q: How do I calculate the slope of a line if I only know one point and the equation of the line?
A: If you know one point and the equation of the line, you can substitute the coordinates of the point into the equation and solve for the slope. Alternatively, you can use the point-slope form of a line, which is y - y1 = m(x - x1), where (x1, y1) is the point and m is the slope.
Q: What is the difference between the slope of a line and the rate of change?
A: The slope of a line and the rate of change are related but not the same thing. The slope of a line is a measure of how steep it is, while the rate of change is a measure of how much the output changes for a given change in the input.
Q: Can the slope of a line be negative?
A: Yes, the slope of a line can be negative. This means that the line is sloping downward from left to right.
Q: Can the slope of a line be zero?
A: Yes, the slope of a line can be zero. This means that the line is horizontal.
Q: Can the slope of a line be undefined?
A: Yes, the slope of a line can be undefined. This means that the line is vertical.
Q: How do I use the slope to find the equation of a line?
A: To find the equation of a line using the slope, you can use the point-slope form of a line, which is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. You can then solve for y to get the equation of the line.
Q: Can I use the slope to find the equation of a line if I only know two points?
A: Yes, you can use the slope to find the equation of a line if you only know two points. You can calculate the slope using the two points and then use the point-slope form of a line to find the equation of the line.
Q: What is the significance of the slope in real-world applications?
A: The slope has many real-world applications, including physics, engineering, economics, and finance. It is used to model the behavior of objects, predict future outcomes, and make informed decisions.
Q: Can I use the slope to predict future outcomes?
A: Yes, you can use the slope to predict future outcomes. By analyzing the slope of a line, you can determine the rate of change and make predictions about future values.
Q: How do I use the slope to make predictions?
A: To use the slope to make predictions, you can analyze the slope of a line and determine the rate of change. You can then use this information to make predictions about future values.
Q: Can I use the slope to model the behavior of objects?
A: Yes, you can use the slope to model the behavior of objects. By analyzing the slope of a line, you can determine the rate of change and model the behavior of objects.
Q: What are some common applications of the slope in real-world scenarios?
A: Some common applications of the slope in real-world scenarios include:
- Modeling the behavior of objects in physics and engineering
- Predicting future outcomes in economics and finance
- Analyzing data in statistics and data analysis
- Making informed decisions in business and management
Conclusion
In this article, we have answered some frequently asked questions about slope to help you better understand this concept. We have discussed the significance of the slope in real-world applications, how to use the slope to find the equation of a line, and how to make predictions using the slope. We hope that this article has been helpful in clarifying any doubts you may have had about slope.
Related Topics
- Calculating the slope of a line given two points
- Understanding the concept of slope
- Applications of slope in mathematics
- Geometry and algebra