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Introduction
In mathematics, finding the equation of a line that passes through a given pair of points is a fundamental concept in algebra and geometry. The slope-intercept form of a linear equation is a powerful tool for representing lines in a two-dimensional coordinate system. In this article, we will explore how to find the equation of a line passing through a given pair of points using the slope-intercept form.
What is the Slope-Intercept Form?
The slope-intercept form of a linear equation is a mathematical expression that represents a line in the form of y = mx + b, where:
- m is the slope of the line
- b is the y-intercept of the line
- x is the independent variable
- y is the dependent variable
The slope-intercept form is a convenient way to represent lines because it allows us to easily identify the slope and y-intercept of the line.
Finding the Slope
To find the equation of a line passing through a given pair of points, we need to find the slope of the line. The slope of a line is a measure of how steep the line is. It can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the coordinates of the two points.
Example
Let's say we have two points: (-5, -1) and (4, 1). We can use the formula above to find the slope of the line passing through these points.
m = (1 - (-1)) / (4 - (-5)) m = (1 + 1) / (4 + 5) m = 2 / 9
So, the slope of the line passing through the points (-5, -1) and (4, 1) is 2/9.
Finding the Y-Intercept
Now that we have the slope, we can find the y-intercept of the line. The y-intercept 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.
Example
Let's say we have the slope (2/9) and one of the points (-5, -1). We can substitute these values into the slope-intercept form of the equation to find the y-intercept.
y = (2/9)x + b -1 = (2/9)(-5) + b -1 = (-10/9) + b b = -1 + 10/9 b = (-9 + 10)/9 b = 1/9
So, the y-intercept of the line passing through the points (-5, -1) and (4, 1) is 1/9.
Writing the Equation in Slope-Intercept Form
Now that we have the slope and y-intercept, we can write the equation of the line in slope-intercept form.
y = (2/9)x + 1/9
This is the equation of the line passing through the points (-5, -1) and (4, 1).
Conclusion
In this article, we learned how to find the equation of a line passing through a given pair of points using the slope-intercept form. We found the slope and y-intercept of the line and used these values to write the equation in slope-intercept form. This is a powerful tool for representing lines in a two-dimensional coordinate system.
Example Problems
Problem 1
Find the equation of the line passing through the points (2, 3) and (4, 5).
Solution
To find the equation of the line passing through the points (2, 3) and (4, 5), we need to find the slope and y-intercept of the line.
m = (5 - 3) / (4 - 2) m = 2 / 2 m = 1
Now that we have the slope, we can find the y-intercept of the line. We can substitute the slope and one of the points into the slope-intercept form of the equation.
y = x + b 3 = 2 + b b = 1
So, the y-intercept of the line passing through the points (2, 3) and (4, 5) is 1.
The equation of the line passing through the points (2, 3) and (4, 5) is:
y = x + 1
Problem 2
Find the equation of the line passing through the points (-3, 2) and (1, 4).
Solution
To find the equation of the line passing through the points (-3, 2) and (1, 4), we need to find the slope and y-intercept of the line.
m = (4 - 2) / (1 - (-3)) m = 2 / 4 m = 1/2
Now that we have the slope, we can find the y-intercept of the line. We can substitute the slope and one of the points into the slope-intercept form of the equation.
y = (1/2)x + b 2 = (1/2)(-3) + b 2 = (-3/2) + b b = 2 + 3/2 b = (4 + 3)/2 b = 7/2
So, the y-intercept of the line passing through the points (-3, 2) and (1, 4) is 7/2.
The equation of the line passing through the points (-3, 2) and (1, 4) is:
y = (1/2)x + 7/2
Final Thoughts
Finding the equation of a line passing through a given pair of points is a fundamental concept in algebra and geometry. The slope-intercept form of a linear equation is a powerful tool for representing lines in a two-dimensional coordinate system. By following the steps outlined in this article, you can find the equation of a line passing through a given pair of points using the slope-intercept form.
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Introduction
In our previous article, we learned how to find the equation of a line passing through a given pair of points using the slope-intercept form. In this article, we will answer some frequently asked questions about finding the equation of a line passing through a given pair of points.
Q&A
Q: What is the slope-intercept form of a linear equation?
A: The slope-intercept form of a linear equation is a mathematical expression that represents a line in the form of y = mx + b, where m is the slope of the line and b is the y-intercept of the line.
Q: How do I find the slope of a line passing through a given pair of points?
A: To find the slope of a line passing through a given pair of points, you can use the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the coordinates of the two points.
Q: How do I find the y-intercept of a line passing through a given pair of points?
A: To find the y-intercept of a line passing through a given pair of points, you can substitute the slope and one of the points into the slope-intercept form of the equation.
Q: What if the slope is zero?
A: If the slope is zero, the line is a horizontal line. In this case, the equation of the line is simply y = b, where b is the y-intercept.
Q: What if the slope is undefined?
A: If the slope is undefined, the line is a vertical line. In this case, the equation of the line is simply x = a, where a is the x-coordinate of the point.
Q: Can I use the slope-intercept form to find the equation of a line passing through three points?
A: No, the slope-intercept form can only be used to find the equation of a line passing through two points. If you have three points, you can use the point-slope form or the two-point form to find the equation of the line.
Q: Can I use the slope-intercept form to find the equation of a line passing through a point and a line?
A: No, the slope-intercept form can only be used to find the equation of a line passing through two points. If you have a point and a line, you can use the point-line form or the two-point form to find the equation of the line.
Q: How do I graph a line using the slope-intercept form?
A: To graph a line using the slope-intercept form, you can use the following steps:
- Find the y-intercept of the line.
- Find the slope of the line.
- Plot the y-intercept on the coordinate plane.
- Use the slope to find another point on the line.
- Plot the second point on the coordinate plane.
- Draw a line through the two points.
Example Problems
Problem 1
Find the equation of the line passing through the points (2, 3) and (4, 5).
Solution
To find the equation of the line passing through the points (2, 3) and (4, 5), we need to find the slope and y-intercept of the line.
m = (5 - 3) / (4 - 2) m = 2 / 2 m = 1
Now that we have the slope, we can find the y-intercept of the line. We can substitute the slope and one of the points into the slope-intercept form of the equation.
y = x + b 3 = 2 + b b = 1
So, the y-intercept of the line passing through the points (2, 3) and (4, 5) is 1.
The equation of the line passing through the points (2, 3) and (4, 5) is:
y = x + 1
Problem 2
Find the equation of the line passing through the points (-3, 2) and (1, 4).
Solution
To find the equation of the line passing through the points (-3, 2) and (1, 4), we need to find the slope and y-intercept of the line.
m = (4 - 2) / (1 - (-3)) m = 2 / 4 m = 1/2
Now that we have the slope, we can find the y-intercept of the line. We can substitute the slope and one of the points into the slope-intercept form of the equation.
y = (1/2)x + b 2 = (1/2)(-3) + b 2 = (-3/2) + b b = 2 + 3/2 b = (4 + 3)/2 b = 7/2
So, the y-intercept of the line passing through the points (-3, 2) and (1, 4) is 7/2.
The equation of the line passing through the points (-3, 2) and (1, 4) is:
y = (1/2)x + 7/2
Final Thoughts
Finding the equation of a line passing through a given pair of points is a fundamental concept in algebra and geometry. The slope-intercept form of a linear equation is a powerful tool for representing lines in a two-dimensional coordinate system. By following the steps outlined in this article, you can find the equation of a line passing through a given pair of points using the slope-intercept form.