Find The Equation Of The Line That Passes Through The Point { (-4, 2)$}$ And Is Parallel To The Line { X - 2y = 6$}$.
===========================================================
Introduction
In mathematics, finding the equation of a line that passes through a given point and is parallel to another line is a fundamental problem in geometry and algebra. This problem involves using the concept of slope and the point-slope form of a linear equation to find the desired equation. In this article, we will discuss how to find the equation of the line that passes through the point (-4, 2) and is parallel to the line x - 2y = 6.
Understanding the Problem
To solve this problem, we need to understand the concept of slope and the point-slope form of a linear equation. The slope of a line is a measure of how steep it is, and it can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are two points on the line.
The point-slope form of a linear equation is given by:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line and m is the slope.
Finding the Slope of the Given Line
The given line is x - 2y = 6. To find the slope of this line, we need to rewrite it in the slope-intercept form, which is:
y = mx + b
where m is the slope and b is the y-intercept.
To rewrite the given line in the slope-intercept form, we need to isolate y. We can do this by subtracting x from both sides of the equation and then dividing both sides by -2.
x - 2y = 6 -2y = -x + 6 y = (1/2)x - 3
Now that we have the slope-intercept form of the given line, we can see that the slope is 1/2.
Finding the Equation of the Parallel Line
Since the line we are looking for is parallel to the given line, it must have the same slope. Therefore, the slope of the line we are looking for is also 1/2.
We are given that the line passes through the point (-4, 2). We can use the point-slope form of a linear equation to find the equation of the line. The point-slope form is given by:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line and m is the slope.
Plugging in the values we know, we get:
y - 2 = (1/2)(x - (-4))
Simplifying this equation, we get:
y - 2 = (1/2)(x + 4) y - 2 = (1/2)x + 2 y = (1/2)x + 4
Conclusion
In this article, we discussed how to find the equation of the line that passes through a given point and is parallel to another line. We used the concept of slope and the point-slope form of a linear equation to find the desired equation. We found that the slope of the given line is 1/2, and therefore the slope of the line we are looking for is also 1/2. We then used the point-slope form to find the equation of the line, which is y = (1/2)x + 4.
Example Problems
Problem 1
Find the equation of the line that passes through the point (2, 3) and is parallel to the line x + 2y = 4.
Solution
To find the equation of the line, we need to find the slope of the given line. We can do this by rewriting the given line in the slope-intercept form.
x + 2y = 4 2y = -x + 4 y = (-1/2)x + 2
Now that we have the slope-intercept form of the given line, we can see that the slope is -1/2.
Since the line we are looking for is parallel to the given line, it must have the same slope. Therefore, the slope of the line we are looking for is also -1/2.
We are given that the line passes through the point (2, 3). We can use the point-slope form of a linear equation to find the equation of the line. The point-slope form is given by:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line and m is the slope.
Plugging in the values we know, we get:
y - 3 = (-1/2)(x - 2)
Simplifying this equation, we get:
y - 3 = (-1/2)x + 1 y = (-1/2)x + 4
Problem 2
Find the equation of the line that passes through the point (-2, 1) and is parallel to the line 2x - 3y = 5.
Solution
To find the equation of the line, we need to find the slope of the given line. We can do this by rewriting the given line in the slope-intercept form.
2x - 3y = 5 -3y = -2x + 5 y = (2/3)x - 5/3
Now that we have the slope-intercept form of the given line, we can see that the slope is 2/3.
Since the line we are looking for is parallel to the given line, it must have the same slope. Therefore, the slope of the line we are looking for is also 2/3.
We are given that the line passes through the point (-2, 1). We can use the point-slope form of a linear equation to find the equation of the line. The point-slope form is given by:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line and m is the slope.
Plugging in the values we know, we get:
y - 1 = (2/3)(x - (-2))
Simplifying this equation, we get:
y - 1 = (2/3)x + 4/3 y = (2/3)x + 7/3
Conclusion
In this article, we discussed how to find the equation of the line that passes through a given point and is parallel to another line. We used the concept of slope and the point-slope form of a linear equation to find the desired equation. We found that the slope of the given line is 1/2, and therefore the slope of the line we are looking for is also 1/2. We then used the point-slope form to find the equation of the line, which is y = (1/2)x + 4.
References
- [1] "Algebra and Trigonometry" by Michael Sullivan
- [2] "Calculus" by Michael Spivak
- [3] "Linear Algebra and Its Applications" by Gilbert Strang
Further Reading
- [1] "The Art of Problem Solving" by Richard Rusczyk
- [2] "Mathematics for the Nonmathematician" by Morris Kline
- [3] "A Course in Mathematics for Students of Physics" by Michael Spivak
===========================================================
Introduction
In our previous article, we discussed how to find the equation of the line that passes through a given point and is parallel to another line. We used the concept of slope and the point-slope form of a linear equation to find the desired equation. In this article, we will answer some frequently asked questions related to this topic.
Q&A
Q: What is the slope of a line that passes through two points (x1, y1) and (x2, y2)?
A: The slope of a line that passes through two points (x1, y1) and (x2, y2) is given by the formula:
m = (y2 - y1) / (x2 - x1)
Q: How do I find the equation of a line that passes through a given point and is parallel to another line?
A: To find the equation of a line that passes through a given point and is parallel to another line, you need to follow these steps:
- Find the slope of the given line.
- Use the point-slope form of a linear equation to find the equation of the line.
Q: What is the point-slope form of a linear equation?
A: The point-slope form of a linear equation is given by:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line and m is the slope.
Q: How do I find the slope of a line from its equation in the slope-intercept form?
A: To find the slope of a line from its equation in the slope-intercept form, you can simply look at the coefficient of x. The slope is the coefficient of x.
Q: Can a line have a slope of zero?
A: Yes, a line can have a slope of zero. This means that the line is horizontal and its equation is of the form y = b, where b is a constant.
Q: Can a line have a slope that is undefined?
A: Yes, a line can have a slope that is undefined. This means that the line is vertical and its equation is of the form x = a, where a is a constant.
Q: How do I find the equation of a line that passes through a given point and is perpendicular to another line?
A: To find the equation of a line that passes through a given point and is perpendicular to another line, you need to follow these steps:
- Find the slope of the given line.
- Find the negative reciprocal of the slope.
- Use the point-slope form of a linear equation to find the equation of the line.
Q: What is the negative reciprocal of a slope?
A: The negative reciprocal of a slope m is given by -1/m.
Q: Can a line have a slope that is equal to its negative reciprocal?
A: No, a line cannot have a slope that is equal to its negative reciprocal. This would mean that the line is perpendicular to itself, which is not possible.
Q: How do I find the equation of a line that passes through two points?
A: To find the equation of a line that passes through two points, you can use the point-slope form of a linear equation. First, find the slope of the line using the formula:
m = (y2 - y1) / (x2 - x1)
Then, use the point-slope form to find the equation of the line.
Q: Can a line have an equation that is of the form x = a?
A: Yes, a line can have an equation that is of the form x = a. This means that the line is vertical and its equation is of the form x = a, where a is a constant.
Q: Can a line have an equation that is of the form y = b?
A: Yes, a line can have an equation that is of the form y = b. This means that the line is horizontal and its equation is of the form y = b, where b is a constant.
Q: How do I find the equation of a line that passes through a given point and has a given slope?
A: To find the equation of a line that passes through a given point and has a given slope, you can use the point-slope form of a linear equation. First, plug in the values of the point and the slope into the point-slope form. Then, simplify the equation to find the desired equation.
Q: Can a line have an equation that is of the form y = mx + b?
A: Yes, a line can have an equation that is of the form y = mx + b. This means that the line is in the slope-intercept form and its equation is of the form y = mx + b, where m is the slope and b is the y-intercept.
Q: How do I find the y-intercept of a line from its equation in the slope-intercept form?
A: To find the y-intercept of a line from its equation in the slope-intercept form, you can simply look at the constant term. The y-intercept is the constant term.
Q: Can a line have a y-intercept that is equal to its slope?
A: No, a line cannot have a y-intercept that is equal to its slope. This would mean that the line is of the form y = y, which is not possible.
Q: How do I find the equation of a line that passes through a given point and has a given y-intercept?
A: To find the equation of a line that passes through a given point and has a given y-intercept, you can use the point-slope form of a linear equation. First, plug in the values of the point and the y-intercept into the point-slope form. Then, simplify the equation to find the desired equation.
Q: Can a line have an equation that is of the form y = mx + b, where m is the slope and b is the y-intercept?
A: Yes, a line can have an equation that is of the form y = mx + b, where m is the slope and b is the y-intercept. This means that the line is in the slope-intercept form and its equation is of the form y = mx + b, where m is the slope and b is the y-intercept.
Q: How do I find the slope of a line from its equation in the slope-intercept form?
A: To find the slope of a line from its equation in the slope-intercept form, you can simply look at the coefficient of x. The slope is the coefficient of x.
Q: Can a line have a slope that is equal to its negative reciprocal?
A: No, a line cannot have a slope that is equal to its negative reciprocal. This would mean that the line is perpendicular to itself, which is not possible.
Q: How do I find the equation of a line that passes through a given point and is parallel to another line?
A: To find the equation of a line that passes through a given point and is parallel to another line, you need to follow these steps:
- Find the slope of the given line.
- Use the point-slope form of a linear equation to find the equation of the line.
Q: Can a line have an equation that is of the form x = a, where a is a constant?
A: Yes, a line can have an equation that is of the form x = a, where a is a constant. This means that the line is vertical and its equation is of the form x = a, where a is a constant.
Q: Can a line have an equation that is of the form y = b, where b is a constant?
A: Yes, a line can have an equation that is of the form y = b, where b is a constant. This means that the line is horizontal and its equation is of the form y = b, where b is a constant.
Q: How do I find the equation of a line that passes through two points?
A: To find the equation of a line that passes through two points, you can use the point-slope form of a linear equation. First, find the slope of the line using the formula:
m = (y2 - y1) / (x2 - x1)
Then, use the point-slope form to find the equation of the line.
Q: Can a line have an equation that is of the form y = mx + b, where m is the slope and b is the y-intercept?
A: Yes, a line can have an equation that is of the form y = mx + b, where m is the slope and b is the y-intercept. This means that the line is in the slope-intercept form and its equation is of the form y = mx + b, where m is the slope and b is the y-intercept.
Q: How do I find the slope of a line from its equation in the slope-intercept form?
A: To find the slope of a line from its equation in the slope-intercept form, you can simply look at the coefficient of x. The slope is the coefficient of x.
Q: Can a line have a slope that is equal to its negative reciprocal?
A: No, a line cannot have a slope that is equal to its negative reciprocal. This would mean that the line is perpendicular to itself, which is not possible.
Q: How do I find the equation of a line that passes through a given point and has a given slope?
A: To find the equation of a line that passes through a given point and has a given slope