Find The Slope Of The Line That Passes Through The Given Points, If Possible. (If An Answer Is Undefined, Enter UNDEFINED.)Points: { (-1, 7)$}$ And { (5, 1)$}$
Introduction
In mathematics, the slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. In this article, we will learn how to find the slope of a line that passes through two given points.
What is the Slope of a Line?
The slope of a line is a measure of its steepness. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The slope is denoted by the letter 'm' and is calculated using the following formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the coordinates of the two points on the line.
Finding the Slope of a Line through Two Points
To find the slope of a line that passes through two points, we can use the formula above. Let's consider the two points (-1, 7) and (5, 1). We can plug these values into the formula to find the slope.
Step 1: Identify the Coordinates of the Two Points
The coordinates of the two points are (-1, 7) and (5, 1). We can identify these points on a coordinate plane.
Step 2: Calculate the Vertical Change (Rise)
The vertical change (rise) is the difference between the y-coordinates of the two points. We can calculate this as follows:
rise = y2 - y1 = 1 - 7 = -6
Step 3: Calculate the Horizontal Change (Run)
The horizontal change (run) is the difference between the x-coordinates of the two points. We can calculate this as follows:
run = x2 - x1 = 5 - (-1) = 6
Step 4: Calculate the Slope
Now that we have the vertical change (rise) and the horizontal change (run), we can calculate the slope using the formula:
m = (y2 - y1) / (x2 - x1) = (-6) / 6 = -1
Conclusion
In this article, we learned how to find the slope of a line that passes through two given points. We used the formula m = (y2 - y1) / (x2 - x1) to calculate the slope of the line that passes through the points (-1, 7) and (5, 1). The slope of the line is -1.
Example Problems
- Find the slope of the line that passes through the points (2, 3) and (4, 5).
- Find the slope of the line that passes through the points (-2, 1) and (3, 4).
Solutions
- The slope of the line that passes through the points (2, 3) and (4, 5) is (5 - 3) / (4 - 2) = 1.
- The slope of the line that passes through the points (-2, 1) and (3, 4) is (4 - 1) / (3 - (-2)) = 3 / 5.
Practice Problems
- Find the slope of the line that passes through the points (1, 2) and (3, 4).
- Find the slope of the line that passes through the points (-3, 2) and (1, 4).
Solutions
- The slope of the line that passes through the points (1, 2) and (3, 4) is (4 - 2) / (3 - 1) = 1.
- The slope of the line that passes through the points (-3, 2) and (1, 4) is (4 - 2) / (1 - (-3)) = 2 / 4 = 1/2.
Conclusion
Q: What is the slope of a line?
A: The slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.
Q: How do I calculate the slope of a line?
A: To calculate the slope of a line, you can use the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the coordinates of the two points on the line.
Q: What if the line is vertical?
A: If the line is vertical, the slope is undefined. This is because the horizontal change (run) is zero, and you cannot divide by zero.
Q: What if the line is horizontal?
A: If the line is horizontal, the slope is zero. This is because the vertical change (rise) is zero, and the ratio of zero to any number is zero.
Q: Can I find the slope of a line if I only know one point?
A: No, you cannot find the slope of a line if you only know one point. You need to know at least two points on the line to calculate the slope.
Q: Can I find the slope of a line if the points are not on a coordinate plane?
A: Yes, you can find the slope of a line even if the points are not on a coordinate plane. You can use the same formula to calculate the slope, but you will need to use the coordinates of the points in a different coordinate system.
Q: How do I use the slope to find the equation of a line?
A: To find the equation of a line, you can use the slope-intercept form of a linear equation:
y = mx + b
where m is the slope and b is the y-intercept. You can plug in the slope and one point on the line to find the equation.
Q: Can I use the slope to find the equation of a line if I only know one point?
A: No, you cannot use the slope to find the equation of a line if you only know one point. You need to know at least two points on the line to find the equation.
Q: Can I use the slope to find the equation of a line if the points are not on a coordinate plane?
A: Yes, you can use the slope to find the equation of a line even if the points are not on a coordinate plane. You can use the same formula to calculate the slope, and then plug in the slope and one point on the line to find the equation.
Q: What is the significance of the slope of a line?
A: The slope of a line is significant because it tells you how steep the line is. It is used in many real-world applications, such as physics, engineering, and economics.
Q: Can I use the slope to solve problems in real-world applications?
A: Yes, you can use the slope to solve problems in real-world applications. For example, you can use the slope to calculate the rate of change of a quantity, or to find the equation of a line that represents a real-world relationship.
Q: What are some common applications of the slope of a line?
A: Some common applications of the slope of a line include:
- Calculating the rate of change of a quantity
- Finding the equation of a line that represents a real-world relationship
- Determining the steepness of a line
- Calculating the slope of a line that passes through two points
Q: Can I use the slope to solve problems in physics?
A: Yes, you can use the slope to solve problems in physics. For example, you can use the slope to calculate the acceleration of an object, or to find the equation of a line that represents a real-world relationship in physics.
Q: Can I use the slope to solve problems in engineering?
A: Yes, you can use the slope to solve problems in engineering. For example, you can use the slope to calculate the stress on a beam, or to find the equation of a line that represents a real-world relationship in engineering.
Q: Can I use the slope to solve problems in economics?
A: Yes, you can use the slope to solve problems in economics. For example, you can use the slope to calculate the rate of change of a quantity, or to find the equation of a line that represents a real-world relationship in economics.