A Line Passes Through The Points (– 13,– 8) And (5,1). Write Its Equation In Slope-intercept Form
Introduction
In mathematics, the slope-intercept form of a line is a fundamental concept that helps us understand the relationship between the slope and the y-intercept of a line. Given two points on a line, we can use the slope-intercept form to find the equation of the line. In this article, we will explore how to find the equation of a line that passes through two points, (–13,–8) and (5,1), in slope-intercept form.
What is Slope-Intercept Form?
The slope-intercept form of a line is a mathematical equation that represents the relationship between the x and y coordinates of a point on the line. It is written in the form y = mx + b, where m is the slope of the line and b is the y-intercept. The slope of a line is a measure of how steep it is, and the y-intercept is the point where the line intersects the y-axis.
Finding the Slope of the Line
To find the equation of the line in slope-intercept form, we need to find the slope of the line first. The slope of a line can be found using the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of the two points on the line. In this case, the two points are (–13,–8) and (5,1).
# Define the coordinates of the two points
x1 = -13
y1 = -8
x2 = 5
y2 = 1
# Calculate the slope of the line
m = (y2 - y1) / (x2 - x1)
print(m)
Calculating the Slope
Using the formula, we can calculate the slope of the line as follows:
m = (1 - (–8)) / (5 - (–13)) m = (1 + 8) / (5 + 13) m = 9 / 18 m = 1/2
Finding the Y-Intercept of the Line
Now that we have the slope of the line, we can find the y-intercept of the line. The y-intercept of a line is the point where the line intersects the y-axis, and it can be found by substituting the slope and one of the points into the slope-intercept form of the equation.
Substituting the Slope and a Point into the Equation
We can substitute the slope (m = 1/2) and one of the points (x = –13, y = –8) into the slope-intercept form of the equation to find the y-intercept.
y = mx + b –8 = (1/2)(–13) + b –8 = –6.5 + b b = –8 + 6.5 b = –1.5
Writing the Equation of the Line in Slope-Intercept Form
Now that we have the slope (m = 1/2) and the y-intercept (b = –1.5), we can write the equation of the line in slope-intercept form.
y = mx + b y = (1/2)x – 1.5
Conclusion
In this article, we have explored how to find the equation of a line that passes through two points, (–13,–8) and (5,1), in slope-intercept form. We have used the slope-intercept form of the equation to find the slope and y-intercept of the line, and we have written the equation of the line in slope-intercept form. This is a fundamental concept in mathematics that helps us understand the relationship between the slope and the y-intercept of a line.
Example Use Cases
The equation of a line in slope-intercept form has many practical applications in real-world scenarios. Here are a few examples:
- Linear Regression: The equation of a line in slope-intercept form is used in linear regression to model the relationship between a dependent variable and one or more independent variables.
- Physics: The equation of a line in slope-intercept form is used in physics to describe the motion of objects under constant acceleration.
- Engineering: The equation of a line in slope-intercept form is used in engineering to design and optimize systems, such as electrical circuits and mechanical systems.
Final Thoughts
In conclusion, the equation of a line in slope-intercept form is a fundamental concept in mathematics that has many practical applications in real-world scenarios. By understanding how to find the equation of a line in slope-intercept form, we can better understand the relationship between the slope and the y-intercept of a line, and we can apply this knowledge to solve problems in a variety of fields.
Introduction
In our previous article, we explored how to find the equation of a line that passes through two points, (–13,–8) and (5,1), in slope-intercept form. We used the slope-intercept form of the equation to find the slope and y-intercept of the line, and we wrote the equation of the line in slope-intercept form. In this article, we will answer some frequently asked questions about finding the equation of a line in slope-intercept form.
Q&A
Q: What is the slope-intercept form of a line?
A: The slope-intercept form of a line is a mathematical equation that represents the relationship between the x and y coordinates of a point on the line. It is written in the form y = mx + b, where m is the slope of the line and b is the y-intercept.
Q: How do I find the slope of a line?
A: To find 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 is the y-intercept of a line?
A: The y-intercept of a line is the point where the line intersects the y-axis. It can be found by substituting the slope and one of the points into the slope-intercept form of the equation.
Q: How do I write the equation of a line in slope-intercept form?
A: To write the equation of a line in slope-intercept form, you need to find the slope and y-intercept of the line. Once you have these values, you can substitute them into the slope-intercept form of the equation, y = mx + b.
Q: What are some real-world applications of the equation of a line in slope-intercept form?
A: The equation of a line in slope-intercept form has many practical applications in real-world scenarios. Some examples include linear regression, physics, and engineering.
Q: Can I use the equation of a line in slope-intercept form to find the equation of a line that passes through three points?
A: No, the equation of a line in slope-intercept form is used to find the equation of a line that passes through two points. If you have three points, you can use the equation of a line in point-slope form to find the equation of the line.
Q: How do I graph a line in slope-intercept form?
A: To graph a line in slope-intercept form, you can use the slope and y-intercept to find two points on the line. Then, you can plot these points on a coordinate plane and draw a line through them.
Example Problems
Problem 1: Find the equation of a line that passes through the points (2,3) and (4,5) in slope-intercept form.
Solution:
m = (5 - 3) / (4 - 2) m = 2 / 2 m = 1
y = mx + b 5 = 1(4) + b 5 = 4 + b b = 1
y = x + 1
Problem 2: Find the equation of a line that passes through the points (–2,4) and (3,–2) in slope-intercept form.
Solution:
m = (–2 - 4) / (3 - (–2)) m = –6 / 5 m = –6/5
y = mx + b –2 = (–6/5)(3) + b –2 = –18/5 + b b = –2 + 18/5 b = (–10 + 18) / 5 b = 8/5
y = (–6/5)x + 8/5
Conclusion
In this article, we have answered some frequently asked questions about finding the equation of a line in slope-intercept form. We have also provided example problems to help you practice finding the equation of a line in slope-intercept form. By understanding how to find the equation of a line in slope-intercept form, you can better understand the relationship between the slope and the y-intercept of a line, and you can apply this knowledge to solve problems in a variety of fields.