Determine The Slope Between Each Set Of Coordinate Pairs. If The Slope Is Undefined, Type DNE In The Answer Box.The Slope Between { (8,-7)$}$ And { (8,-2)$}$ Is { \square$}$The Slope Between { (1,6)$}$ And
Introduction
In mathematics, the slope between two points on a coordinate plane is a fundamental concept used to describe the steepness of a line. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between the two points. In this article, we will explore how to determine the slope between each set of coordinate pairs, including cases where the slope is undefined.
What is Slope?
Slope, also known as gradient, is a measure of how steep a line is. It is calculated using the formula:
m = (y2 - y1) / (x2 - x1)
where m is the slope, and (x1, y1) and (x2, y2) are the coordinates of the two points.
Calculating Slope
To calculate the slope between two points, we need to follow these steps:
- Identify the coordinates of the two points.
- Calculate the vertical change (rise) between the two points.
- Calculate the horizontal change (run) between the two points.
- Divide the vertical change by the horizontal change to get the slope.
Example 1: Calculating Slope Between Two Points
Let's calculate the slope between the points (8, -7) and (8, -2).
- Identify the coordinates of the two points: (8, -7) and (8, -2).
- Calculate the vertical change (rise): -2 - (-7) = 5.
- Calculate the horizontal change (run): 8 - 8 = 0.
- Since the horizontal change is 0, the slope is undefined.
Example 2: Calculating Slope Between Two Points
Let's calculate the slope between the points (1, 6) and (3, 8).
- Identify the coordinates of the two points: (1, 6) and (3, 8).
- Calculate the vertical change (rise): 8 - 6 = 2.
- Calculate the horizontal change (run): 3 - 1 = 2.
- Divide the vertical change by the horizontal change to get the slope: 2 / 2 = 1.
Example 3: Calculating Slope Between Two Points
Let's calculate the slope between the points (2, 3) and (4, 5).
- Identify the coordinates of the two points: (2, 3) and (4, 5).
- Calculate the vertical change (rise): 5 - 3 = 2.
- Calculate the horizontal change (run): 4 - 2 = 2.
- Divide the vertical change by the horizontal change to get the slope: 2 / 2 = 1.
Conclusion
In conclusion, determining the slope between each set of coordinate pairs is a crucial concept in mathematics. By following the steps outlined in this article, you can calculate the slope between any two points on a coordinate plane. Remember to check if the horizontal change is 0, as this will result in an undefined slope.
Common Mistakes to Avoid
When calculating the slope between two points, there are several common mistakes to avoid:
- Failing to identify the coordinates of the two points.
- Calculating the vertical change incorrectly.
- Calculating the horizontal change incorrectly.
- Dividing by zero (resulting in an undefined slope).
Tips and Tricks
Here are some tips and tricks to help you calculate the slope between two points:
- Use a ruler or a straightedge to draw a line between the two points.
- Calculate the vertical change and horizontal change separately.
- Check if the horizontal change is 0 before calculating the slope.
- Use a calculator to simplify the calculation.
Practice Problems
Try calculating the slope between the following sets of coordinate pairs:
- (2, 4) and (4, 6)
- (1, 2) and (3, 4)
- (5, 7) and (8, 9)
Answer Key
- (2, 4) and (4, 6): 2 / 2 = 1
- (1, 2) and (3, 4): 2 / 2 = 1
- (5, 7) and (8, 9): 2 / 3
Final Thoughts
Introduction
In our previous article, we explored how to determine the slope between each set of coordinate pairs. In this article, we will answer some frequently asked questions about slope and provide additional examples to help you understand the concept better.
Q: What is the slope between two points if the horizontal change is 0?
A: If the horizontal change is 0, the slope is undefined. This is because division by zero is undefined in mathematics.
Q: How do I calculate the slope between two points if the vertical change is 0?
A: If the vertical change is 0, the slope is 0. This is because the ratio of 0 to any number is 0.
Q: Can the slope between two points be a fraction?
A: Yes, the slope between two points can be a fraction. For example, if the vertical change is 3 and the horizontal change is 2, the slope is 3/2.
Q: How do I determine if the slope between two points is positive or negative?
A: To determine if the slope between two points is positive or negative, you need to look at the signs of the vertical and horizontal changes. If the vertical change and horizontal change have the same sign (both positive or both negative), the slope is positive. If the vertical change and horizontal change have opposite signs (one positive and one negative), the slope is negative.
Q: Can the slope between two points be a decimal?
A: Yes, the slope between two points can be a decimal. For example, if the vertical change is 3.5 and the horizontal change is 2, the slope is 3.5/2 = 1.75.
Q: How do I calculate the slope between two points if the coordinates are given in a different order?
A: To calculate the slope between two points if the coordinates are given in a different order, you need to swap the x and y coordinates of one of the points. For example, if the coordinates are (x1, y1) and (x2, y2), you can swap the x and y coordinates of the first point to get (y1, x1) and (x2, y2).
Q: Can the slope between two points be a negative fraction?
A: Yes, the slope between two points can be a negative fraction. For example, if the vertical change is -3 and the horizontal change is 2, the slope is -3/2.
Q: How do I determine if the slope between two points is an integer?
A: To determine if the slope between two points is an integer, you need to check if the ratio of the vertical change to the horizontal change is an integer. If the ratio is an integer, the slope is an integer.
Q: Can the slope between two points be a negative decimal?
A: Yes, the slope between two points can be a negative decimal. For example, if the vertical change is -3.5 and the horizontal change is 2, the slope is -3.5/2 = -1.75.
Q: How do I calculate the slope between two points if the coordinates are given in a different unit?
A: To calculate the slope between two points if the coordinates are given in a different unit, you need to convert the coordinates to the same unit. For example, if the coordinates are given in feet and you want to calculate the slope in inches, you need to convert the coordinates from feet to inches.
Conclusion
In conclusion, calculating the slope between two points is a fundamental concept in mathematics. By following the steps outlined in this article and answering the frequently asked questions, you can determine the slope between any two points on a coordinate plane. Remember to check if the horizontal change is 0, as this will result in an undefined slope. With practice and patience, you will become proficient in calculating the slope between two points.