Write The Equation Of The Line That Passes Through The Points { (7, -4)$}$ And { (-1, 3)$}$, First In Point-slope Form, And Then In Slope-intercept Form.1. The Slope Of The Line Is { \square$}$.2. Using The Point
Introduction
In mathematics, the equation of a line is a fundamental concept that is used to describe the relationship between two variables. Given two points on a line, we can use the concept of slope to find the equation of the line. In this article, we will explore how to write the equation of a line that passes through the points {(7, -4)$}$ and {(-1, 3)$}$, first in point-slope form, and then in slope-intercept form.
Calculating the Slope
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. To calculate the slope, we can use the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the coordinates of the two points.
In this case, the two points are (7, -4) and (-1, 3). Plugging these values into the formula, we get:
m = (3 - (-4)) / (-1 - 7) m = (3 + 4) / (-1 - 7) m = 7 / -8 m = -7/8
The Slope of the Line
So, the slope of the line is -7/8.
Writing the Equation in Point-Slope Form
The point-slope form of a line is given by the equation:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line and m is the slope.
In this case, we can use the point (7, -4) and the slope m = -7/8 to write the equation of the line in point-slope form:
y - (-4) = -7/8(x - 7) y + 4 = -7/8(x - 7)
Simplifying the Equation
To simplify the equation, we can multiply both sides by 8 to eliminate the fraction:
8(y + 4) = -7(x - 7) 8y + 32 = -7x + 49
Writing the Equation in Slope-Intercept Form
The slope-intercept form of a line is given by the equation:
y = mx + b
where m is the slope and b is the y-intercept.
To write the equation in slope-intercept form, we can rearrange the equation we obtained in point-slope form:
8y + 32 = -7x + 49 8y = -7x + 17 y = (-7/8)x + 17/8
The Equation of the Line
So, the equation of the line that passes through the points {(7, -4)$}$ and {(-1, 3)$}$ is:
y = (-7/8)x + 17/8
Conclusion
In this article, we have seen how to write the equation of a line that passes through two points. We have used the concept of slope to find the equation of the line in both point-slope form and slope-intercept form. The equation of the line is a powerful tool that can be used to describe the relationship between two variables.
Key Takeaways
- The slope of a line is a measure of how steep it is.
- The point-slope form of a line is given by the equation y - y1 = m(x - x1).
- The slope-intercept form of a line is given by the equation y = mx + b.
- The equation of a line can be written in both point-slope form and slope-intercept form.
Further Reading
If you want to learn more about the equation of a line, I recommend checking out the following resources:
- Khan Academy: Equation of a Line
- Math Is Fun: Equation of a Line
- Wolfram MathWorld: Equation of a Line
Frequently Asked Questions: The Equation of a Line =====================================================
Q: What is the equation of a line?
A: The equation of a line is a mathematical expression that describes the relationship between two variables, typically x and y. It is a way to represent the graph of a line on a coordinate plane.
Q: How do I find the equation of a line?
A: To find the equation of a line, you need to know two points on the line. You can use the concept of slope to find the equation of the line in both point-slope form and slope-intercept form.
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.
Q: What is the point-slope form of a line?
A: The point-slope form of a line is given by the equation:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line and m is the slope.
Q: What is the slope-intercept form of a line?
A: The slope-intercept form of a line is given by the equation:
y = mx + b
where m is the slope and b is the y-intercept.
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 can rearrange the equation you obtained in point-slope form.
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 is the value of y when x is equal to 0.
Q: How do I find the y-intercept of a line?
A: To find the y-intercept of a line, you can set x equal to 0 in the equation of the line and solve for y.
Q: What are some common applications of the equation of a line?
A: The equation of a line has many practical applications in fields such as physics, engineering, and economics. Some common applications include:
- Modeling the motion of objects
- Describing the relationship between two variables
- Finding the equation of a line that passes through two points
- Solving systems of linear equations
Q: What are some common mistakes to avoid when working with the equation of a line?
A: Some common mistakes to avoid when working with the equation of a line include:
- Not using the correct formula for the slope
- Not using the correct formula for the equation of a line
- Not checking the units of the variables
- Not checking the domain and range of the function
Conclusion
In this article, we have answered some frequently asked questions about the equation of a line. We have covered topics such as the slope of a line, the point-slope form of a line, and the slope-intercept form of a line. We have also discussed some common applications and mistakes to avoid when working with the equation of a line.