What Is An Equation Of The Line That Is Parallel To $y = -5x + 6$ And Passes Through The Point $(-4, -1$\]?A. $y = -5x + 21$B. $y = -5x - 21$C. $y = -5x - 19$D. $y = -5x + 19$
Introduction
In mathematics, the concept of parallel lines is a fundamental idea in geometry and algebra. Two lines are said to be parallel if they lie in the same plane and never intersect, no matter how far they are extended. In this article, we will explore the concept of parallel lines and how to find the equation of a line that is parallel to a given line and passes through a specific point.
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 are always the same distance apart and never touch each other. The concept of parallel lines is often used in geometry and algebra to solve problems involving lines and angles.
Slope of a Line
The slope of a line is a measure of how steep it is. It is calculated by dividing the vertical change (rise) by the horizontal change (run). The slope of a line can be positive, negative, or zero. A positive slope indicates that the line is rising from left to right, while a negative slope indicates that the line is falling from left to right. A slope of zero indicates that the line is horizontal.
Finding the Equation of a Line
To find the equation of a line, we need to know its slope and a point on the line. The equation of a line can be written in the form y = mx + b, where m is the slope and b is the y-intercept. The y-intercept is the point where the line intersects the y-axis.
Parallel Lines and Slope
Parallel lines have the same slope. This means that if we know the slope of one line, we can find the slope of any other line that is parallel to it. In the case of the given line y = -5x + 6, the slope is -5. Therefore, any line that is parallel to this line will also have a slope of -5.
Finding the Equation of a Line that is Parallel to and Passes Through the Point
To find the equation of a line that is parallel to y = -5x + 6 and passes through the point (-4, -1), we need to use the point-slope form of a line. The point-slope form of a line is given by y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.
Step 1: Identify the Slope
The slope of the given line is -5. Since the line we are looking for is parallel to this line, it will also have a slope of -5.
Step 2: Identify the Point
The point we are given is (-4, -1). This point lies on the line we are trying to find.
Step 3: Use the Point-Slope Form
Using the point-slope form of a line, we can write the equation of the line as y - (-1) = -5(x - (-4)). Simplifying this equation, we get y + 1 = -5(x + 4).
Step 4: Simplify the Equation
Expanding the equation, we get y + 1 = -5x - 20. Subtracting 1 from both sides, we get y = -5x - 21.
Conclusion
In this article, we explored the concept of parallel lines and how to find the equation of a line that is parallel to a given line and passes through a specific point. We used the point-slope form of a line to find the equation of the line that is parallel to y = -5x + 6 and passes through the point (-4, -1). The equation of this line is y = -5x - 21.
Final Answer
The final answer is y = -5x - 21.
Discussion
The discussion of this problem involves understanding the concept of parallel lines and how to find the equation of a line that is parallel to a given line and passes through a specific point. The point-slope form of a line is a useful tool for solving problems involving lines and angles.
Related Problems
- Find the equation of a line that is parallel to y = 2x + 3 and passes through the point (1, 5).
- Find the equation of a line that is parallel to y = -3x + 2 and passes through the point (-2, 4).
- Find the equation of a line that is parallel to y = x - 1 and passes through the point (3, 2).
Solutions to Related Problems
- The equation of the line that is parallel to y = 2x + 3 and passes through the point (1, 5) is y = 2x + 11.
- The equation of the line that is parallel to y = -3x + 2 and passes through the point (-2, 4) is y = -3x + 14.
- The equation of the line that is parallel to y = x - 1 and passes through the point (3, 2) is y = x + 1.
Conclusion
In conclusion, the concept of parallel lines is a fundamental idea in geometry and algebra. Finding the equation of a line that is parallel to a given line and passes through a specific point involves using the point-slope form of a line. The point-slope form of a line is a useful tool for solving problems involving lines and angles.
Introduction
In our previous article, we explored the concept of parallel lines and how to find the equation of a line that is parallel to a given line and passes through a specific point. In this article, we will answer some frequently asked questions about parallel lines and equations of lines.
Q: What is the difference between parallel lines and perpendicular lines?
A: Parallel lines are lines that lie in the same plane and never intersect, no matter how far they are extended. Perpendicular lines, on the other hand, are lines that intersect at a 90-degree angle.
Q: How do I find the equation of a line that is parallel to a given line and passes through a specific point?
A: To find the equation of a line that is parallel to a given line and passes through a specific point, you can use the point-slope form of a line. The point-slope form of a line is given by y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.
Q: What is the slope of a line that is parallel to y = 2x + 3?
A: The slope of a line that is parallel to y = 2x + 3 is also 2. This is because parallel lines have the same slope.
Q: How do I find the equation of a line that is perpendicular to a given line and passes through a specific point?
A: To find the equation of a line that is perpendicular to a given line and passes through a specific point, you can use the point-slope form of a line. However, you will need to find the slope of the perpendicular line first. The slope of a line that is perpendicular to a given line is the negative reciprocal of the slope of the given line.
Q: What is the equation of a line that is perpendicular to y = 3x - 2 and passes through the point (1, 4)?
A: To find the equation of a line that is perpendicular to y = 3x - 2 and passes through the point (1, 4), we need to find the slope of the perpendicular line first. The slope of the perpendicular line is the negative reciprocal of the slope of the given line, which is -1/3. Using the point-slope form of a line, we can write the equation of the line as y - 4 = (-1/3)(x - 1). Simplifying this equation, we get y = (-1/3)x + 5/3.
Q: How do I find the equation of a line that is parallel to a given line and passes through two specific points?
A: To find the equation of a line that is parallel to a given line and passes through two specific points, you can use the two-point form of a line. The two-point form of a line is given by y - y1 = m(x - x1), where m is the slope and (x1, y1) and (x2, y2) are two points on the line.
Q: What is the equation of a line that is parallel to y = 2x + 3 and passes through the points (1, 5) and (2, 7)?
A: To find the equation of a line that is parallel to y = 2x + 3 and passes through the points (1, 5) and (2, 7), we need to find the slope of the line first. The slope of the line is 2. Using the two-point form of a line, we can write the equation of the line as y - 5 = 2(x - 1). Simplifying this equation, we get y = 2x + 3.
Conclusion
In conclusion, the concept of parallel lines and equations of lines is a fundamental idea in geometry and algebra. By understanding the concept of parallel lines and how to find the equation of a line that is parallel to a given line and passes through a specific point, you can solve a wide range of problems involving lines and angles.
Final Answer
The final answer is that the equation of a line that is parallel to y = -5x + 6 and passes through the point (-4, -1) is y = -5x - 21.
Discussion
The discussion of this problem involves understanding the concept of parallel lines and how to find the equation of a line that is parallel to a given line and passes through a specific point. The point-slope form of a line is a useful tool for solving problems involving lines and angles.
Related Problems
- Find the equation of a line that is parallel to y = 2x + 3 and passes through the points (1, 5) and (2, 7).
- Find the equation of a line that is parallel to y = -3x + 2 and passes through the points (-2, 4) and (1, 6).
- Find the equation of a line that is parallel to y = x - 1 and passes through the points (3, 2) and (4, 5).
Solutions to Related Problems
- The equation of the line that is parallel to y = 2x + 3 and passes through the points (1, 5) and (2, 7) is y = 2x + 3.
- The equation of the line that is parallel to y = -3x + 2 and passes through the points (-2, 4) and (1, 6) is y = -3x + 10.
- The equation of the line that is parallel to y = x - 1 and passes through the points (3, 2) and (4, 5) is y = x + 1.
Conclusion
In conclusion, the concept of parallel lines and equations of lines is a fundamental idea in geometry and algebra. By understanding the concept of parallel lines and how to find the equation of a line that is parallel to a given line and passes through a specific point, you can solve a wide range of problems involving lines and angles.