There Is A Line That Includes The Point { (-8, -2)$}$ And Has A Slope Of { -\frac{1}{6}$}$. What Is Its Equation In Point-slope Form?Use The Specified Point In Your Equation. Write Your Answer Using Integers, Proper Fractions, And
Introduction
When it comes to graphing lines, there are several ways to represent their equations. One of the most common forms is the point-slope form, which is used to describe a line that passes through a specific point and has a given slope. In this article, we will explore how to write the equation of a line in point-slope form, given a specific point and slope.
Understanding 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 of the line. This form is useful because it allows us to easily find the equation of a line that passes through a specific point and has a given slope.
Finding the Equation of a Line in Point-Slope Form
To find the equation of a line in point-slope form, we need to know the coordinates of a point on the line and the slope of the line. In this case, we are given the point (-8, -2) and the slope -1/6.
Step 1: Write the Point-Slope Form Equation
The point-slope form equation is given by:
y - y1 = m(x - x1)
We can plug in the values of the point and the slope into this equation:
y - (-2) = (-1/6)(x - (-8))
Step 2: Simplify the Equation
To simplify the equation, we can start by multiplying both sides by 6 to eliminate the fraction:
6(y - (-2)) = 6((-1/6)(x - (-8)))
This simplifies to:
6y + 12 = -x + 8
Step 3: Rearrange the Equation
To put the equation in the standard form, we can rearrange it to isolate the y-term:
6y = -x - 4
Step 4: Write the Equation in Point-Slope Form
Now that we have the equation in the standard form, we can write it in point-slope form by adding the y-term to both sides:
y = (-1/6)x - 2/3
This is the equation of the line in point-slope form.
Conclusion
In this article, we explored how to write the equation of a line in point-slope form, given a specific point and slope. We used the point (-8, -2) and the slope -1/6 to find the equation of the line in point-slope form. The final equation is:
y = (-1/6)x - 2/3
This equation represents the line that passes through the point (-8, -2) and has a slope of -1/6.
Example Problems
Problem 1
Find the equation of a line in point-slope form that passes through the point (2, 3) and has a slope of 1/2.
Solution
To find the equation of the line in point-slope form, we can use the point-slope form equation:
y - y1 = m(x - x1)
We can plug in the values of the point and the slope into this equation:
y - 3 = (1/2)(x - 2)
To simplify the equation, we can start by multiplying both sides by 2 to eliminate the fraction:
2(y - 3) = 2((1/2)(x - 2))
This simplifies to:
2y - 6 = x - 2
To put the equation in the standard form, we can rearrange it to isolate the y-term:
2y = x + 4
To write the equation in point-slope form, we can add the y-term to both sides:
y = (1/2)x + 2
This is the equation of the line in point-slope form.
Problem 2
Find the equation of a line in point-slope form that passes through the point (-4, 1) and has a slope of -3/4.
Solution
To find the equation of the line in point-slope form, we can use the point-slope form equation:
y - y1 = m(x - x1)
We can plug in the values of the point and the slope into this equation:
y - 1 = (-3/4)(x - (-4))
To simplify the equation, we can start by multiplying both sides by 4 to eliminate the fraction:
4(y - 1) = 4((-3/4)(x - (-4)))
This simplifies to:
4y - 4 = -3x - 12
To put the equation in the standard form, we can rearrange it to isolate the y-term:
4y = 3x + 8
To write the equation in point-slope form, we can add the y-term to both sides:
y = (3/4)x + 2
This is the equation of the line in point-slope form.
Final Answer
The final answer is: y = (-1/6)x - 2/3
Introduction
In our previous article, we explored how to write the equation of a line in point-slope form, given a specific point and slope. In this article, we will answer some frequently asked questions about the point-slope form of a line.
Q1: What is the point-slope form of a line?
A1: 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 of the line.
Q2: How do I find the equation of a line in point-slope form?
A2: To find the equation of a line in point-slope form, you need to know the coordinates of a point on the line and the slope of the line. You can use the point-slope form equation:
y - y1 = m(x - x1)
and plug in the values of the point and the slope.
Q3: What if the slope is a fraction?
A3: If the slope is a fraction, you can multiply both sides of the equation by the denominator to eliminate the fraction. For example, if the slope is 1/2, you can multiply both sides by 2 to get:
2(y - y1) = 2(m)(x - x1)
Q4: Can I use the point-slope form to find the equation of a horizontal line?
A4: Yes, you can use the point-slope form to find the equation of a horizontal line. A horizontal line has a slope of 0, so you can plug in 0 for the slope in the point-slope form equation:
y - y1 = 0(x - x1)
This simplifies to:
y = y1
Q5: Can I use the point-slope form to find the equation of a vertical line?
A5: Yes, you can use the point-slope form to find the equation of a vertical line. A vertical line has an undefined slope, so you can plug in a value of infinity for the slope in the point-slope form equation:
y - y1 = ∞(x - x1)
This simplifies to:
x = x1
Q6: How do I convert the point-slope form to the standard form?
A6: To convert the point-slope form to the standard form, you can rearrange the equation to isolate the y-term. For example, if the point-slope form equation is:
y - 2 = (1/2)(x - 3)
you can add 2 to both sides to get:
y = (1/2)x + 5/2
Q7: Can I use the point-slope form to find the equation of a line that passes through two points?
A7: Yes, you can use the point-slope form to find the equation of a line that passes through two points. You can use the two points to find the slope of the line, and then use the point-slope form equation to find the equation of the line.
Q8: How do I find the slope of a line given two points?
A8: To find the slope of a line given two points, you can use the slope formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the two points.
Q9: Can I use the point-slope form to find the equation of a line that passes through a point and has a given slope?
A9: Yes, you can use the point-slope form to find the equation of a line that passes through a point and has a given slope. You can plug in the values of the point and the slope into the point-slope form equation:
y - y1 = m(x - x1)
Q10: How do I know if a line is in point-slope form?
A10: A line is in point-slope form if it can be written in the form:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line, and m is the slope of the line.
Final Answer
The final answer is: 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 of the line.