The Given Line Passes Through The Points { (0, -3)$}$ And { (2, 3)$}$.What Is The Equation, In Point-slope Form, Of The Line That Is Parallel To The Given Line And Passes Through The Point { (-1, -1)$} ? A . \[ ?A. \[ ? A . \[ Y + 1 =
Understanding the Problem
The problem requires finding the equation of a line that is parallel to a given line and passes through a specific point. To solve this problem, we need to understand the concept of parallel lines and how to find their equations.
What are Parallel Lines?
Parallel lines are lines that lie in the same plane and never intersect, no matter how far they are extended. In other words, parallel lines are lines that have the same slope but different y-intercepts.
Finding the Slope of the Given Line
To find the slope of the given line, we can use the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are two points on the line.
In this case, the two points are (0, -3) and (2, 3). Plugging these values into the formula, we get:
m = (3 - (-3)) / (2 - 0) m = 6 / 2 m = 3
So, the slope of the given line is 3.
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, which is 3. We are also given a point through which the line passes, which is (-1, -1).
To find the equation of the line in point-slope form, we can use the formula:
y - y1 = m(x - x1)
where (x1, y1) is the point through which the line passes, and m is the slope of the line.
Plugging in the values, we get:
y - (-1) = 3(x - (-1)) y + 1 = 3(x + 1)
Simplifying the Equation
To simplify the equation, we can expand the right-hand side:
y + 1 = 3x + 3
The Final Answer
Therefore, the equation of the line that is parallel to the given line and passes through the point (-1, -1) is:
y + 1 = 3x + 3
This is the equation in point-slope form.
Conclusion
In this problem, we found the equation of a line that is parallel to a given line and passes through a specific point. We used the concept of parallel lines and the formula for finding the slope of a line to solve the problem. We also used the point-slope form of a line to find the equation of the parallel line.
Key Takeaways
- Parallel lines have the same slope but different y-intercepts.
- The slope of a line can be found using the formula m = (y2 - y1) / (x2 - x1).
- The equation of a line in point-slope form is given by y - y1 = m(x - x1).
- To find the equation of a parallel line, we need to find the slope of the given line and use the point-slope form of a line.
Additional Examples
- Find the equation of a line that is parallel to the line y = 2x + 1 and passes through the point (0, 3).
- Find the equation of a line that is parallel to the line y = -x + 2 and passes through the point (1, 0).
Solutions to Additional Examples
- The equation of the line that is parallel to the line y = 2x + 1 and passes through the point (0, 3) is y - 3 = 2(x - 0).
- The equation of the line that is parallel to the line y = -x + 2 and passes through the point (1, 0) is y - 0 = -1(x - 1).
Conclusion
Q: What is the definition of parallel lines?
A: Parallel lines are lines that lie in the same plane and never intersect, no matter how far they are extended. In other words, parallel lines are lines that have the same slope but different y-intercepts.
Q: How do I find the slope of a line?
A: To find the slope of a line, you can use the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are two points on the line.
Q: What is the point-slope form of a line?
A: The point-slope form of a line is given by:
y - y1 = m(x - x1)
where (x1, y1) is the point through which the line passes, and m is the slope of the line.
Q: How do I find the equation of a parallel line?
A: To find the equation of a parallel line, you need to find the slope of the given line and use the point-slope form of a line. Since the line you are looking for is parallel to the given line, it must have the same slope.
Q: What is the equation of a line that is parallel to the line y = 2x + 1 and passes through the point (0, 3)?
A: To find the equation of the line that is parallel to the line y = 2x + 1 and passes through the point (0, 3), you need to find the slope of the given line, which is 2. Then, you can use the point-slope form of a line to find the equation of the parallel line:
y - 3 = 2(x - 0)
Q: What is the equation of a line that is parallel to the line y = -x + 2 and passes through the point (1, 0)?
A: To find the equation of the line that is parallel to the line y = -x + 2 and passes through the point (1, 0), you need to find the slope of the given line, which is -1. Then, you can use the point-slope form of a line to find the equation of the parallel line:
y - 0 = -1(x - 1)
Q: How do I know if two lines are parallel?
A: Two lines are parallel if they have the same slope but different y-intercepts.
Q: Can two lines be parallel if they have the same y-intercept?
A: No, two lines cannot be parallel if they have the same y-intercept. If two lines have the same y-intercept, they are the same line.
Q: Can a line be parallel to itself?
A: No, a line cannot be parallel to itself. A line is always parallel to itself, but this is a trivial case and not what we typically mean by "parallel lines".
Q: Can a line be parallel to a line that is not a straight line?
A: No, a line cannot be parallel to a line that is not a straight line. Parallel lines must be straight lines.
Conclusion
In this Q&A article, we answered some common questions about the equation of a parallel line. We covered topics such as the definition of parallel lines, how to find the slope of a line, and how to find the equation of a parallel line. We also provided examples of how to find the equation of a parallel line and how to determine if two lines are parallel.