Write The Equation In Point-slope Form Of The Line That Passes Through The Given Point And Has The Given Slope: Point: (4, -7), Slope: M = − 1 4 M = -\frac{1}{4} M = − 4 1 A. Y + 7 = − 1 4 ( X − 4 Y + 7 = -\frac{1}{4}(x - 4 Y + 7 = − 4 1 ( X − 4 ] B. Y − 4 = − 1 4 ( X + 7 Y - 4 = -\frac{1}{4}(x + 7 Y − 4 = − 4 1 ( X + 7 ] C.
Introduction
In mathematics, the point-slope form of a line is a way to express the equation of a line that passes through a given point and has a given slope. The point-slope form is given by the equation:
y - y1 = m(x - x1)
where (x1, y1) is the given point and m is the slope of the line. In this article, we will use the point-slope form to find the equation of a line that passes through the point (4, -7) and has a slope of m = -1/4.
Step 1: Identify the Given Point and Slope
The given point is (4, -7) and the slope is m = -1/4. We will use these values to find the equation of the line in point-slope form.
Step 2: Plug in the Values into the Point-Slope Form
To find the equation of the line in point-slope form, we will plug in the values of the given point and slope into the equation:
y - y1 = m(x - x1)
Substituting the values, we get:
y - (-7) = -1/4(x - 4)
Step 3: Simplify the Equation
To simplify the equation, we will start by combining the constants on the left-hand side:
y + 7 = -1/4(x - 4)
Next, we will distribute the negative sign to the terms inside the parentheses:
y + 7 = -1/4x + 1
Step 4: Write the Equation in Point-Slope Form
The equation is now in point-slope form:
y + 7 = -1/4(x - 4)
This is the equation of the line that passes through the point (4, -7) and has a slope of m = -1/4.
Conclusion
In this article, we used the point-slope form to find the equation of a line that passes through the point (4, -7) and has a slope of m = -1/4. We plugged in the values into the point-slope form, simplified the equation, and wrote the final equation in point-slope form.
Answer
The correct answer is:
A. y + 7 = -1/4(x - 4)
Discussion
The point-slope form is a useful way to express the equation of a line that passes through a given point and has a given slope. It is a fundamental concept in mathematics and is used in a variety of applications, including algebra, geometry, and calculus.
Example Problems
- Find the equation of the line that passes through the point (2, 3) and has a slope of m = 2.
- Find the equation of the line that passes through the point (-1, 4) and has a slope of m = -3.
- Find the equation of the line that passes through the point (5, -2) and has a slope of m = 1/2.
Solutions
- y - 3 = 2(x - 2)
- y - 4 = -3(x + 1)
- y + 2 = 1/2(x - 5)
Tips and Tricks
- Make sure to plug in the values correctly into the point-slope form.
- Simplify the equation by combining like terms.
- Write the final equation in point-slope form.
Common Mistakes
- Plugging in the values incorrectly into the point-slope form.
- Not simplifying the equation.
- Writing the final equation in the wrong form.
Conclusion
Q: What is the point-slope form of a line?
A: The point-slope form of a line is a way to express the equation of a line that passes through a given point and has a given slope. It is given by the equation:
y - y1 = m(x - x1)
where (x1, y1) is the given point and m is the slope of the line.
Q: How do I find the equation of a line in point-slope form?
A: To find the equation of a line in point-slope form, you need to plug in the values of the given point and slope into the equation:
y - y1 = m(x - x1)
Then, simplify the equation by combining like terms.
Q: What is the difference between the point-slope form and the slope-intercept form?
A: The point-slope form and the slope-intercept form are two different ways to express the equation of a line. The point-slope form is given by the equation:
y - y1 = m(x - x1)
while the slope-intercept form is given by the equation:
y = mx + b
where m is the slope and b is the y-intercept.
Q: How do I convert the point-slope form to the slope-intercept form?
A: To convert the point-slope form to the slope-intercept form, you need to isolate y on one side of the equation. This can be done by adding y1 to both sides of the equation and then dividing both sides by m.
Q: What is the significance of the point-slope form?
A: The point-slope form is a useful way to express the equation of a line that passes through a given point and has a given slope. It is a fundamental concept in mathematics and is used in a variety of applications, including algebra, geometry, and calculus.
Q: Can I use the point-slope form to find the equation of a line that passes through two points?
A: Yes, you can use the point-slope form to find the equation of a line that passes through two points. However, you need to find the slope of the line first using the two points.
Q: How do I find the slope of a line that passes through two points?
A: To find the slope of a line that passes through two points, you can use the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the two points.
Q: Can I use the point-slope form to find the equation of a line that is parallel to another line?
A: Yes, you can use the point-slope form to find the equation of a line that is parallel to another line. However, you need to find the slope of the parallel line first.
Q: How do I find the slope of a line that is parallel to another line?
A: To find the slope of a line that is parallel to another line, you can use the fact that parallel lines have the same slope. Therefore, if you know the slope of one line, you can use it to find the slope of the parallel line.
Q: Can I use the point-slope form to find the equation of a line that is perpendicular to another line?
A: Yes, you can use the point-slope form to find the equation of a line that is perpendicular to another line. However, you need to find the slope of the perpendicular line first.
Q: How do I find the slope of a line that is perpendicular to another line?
A: To find the slope of a line that is perpendicular to another line, you can use the fact that perpendicular lines have slopes that are negative reciprocals of each other. Therefore, if you know the slope of one line, you can use it to find the slope of the perpendicular line.
Conclusion
In this article, we answered some common questions about the point-slope form of a line. We discussed how to find the equation of a line in point-slope form, how to convert the point-slope form to the slope-intercept form, and how to find the slope of a line that passes through two points or is parallel or perpendicular to another line. We also provided some tips and tricks for using the point-slope form.