Find The Slope-intercept Form Of The Equation Of The Line That Passes Through \[$(4, -1)\$\] And Is Parallel To \[$y - \frac{19}{6} = 2\left(x + \frac{7}{2}\right)\$\].
Understanding the Problem
To find the slope-intercept form of the equation of a line that passes through a given point and is parallel to another line, we need to understand the concept of parallel lines and the slope-intercept form of a line. Parallel lines are lines that lie in the same plane and never intersect, which means they have the same slope but different y-intercepts.
The Slope-Intercept Form of a Line
The slope-intercept form of a line is given by the equation y = mx + b, where m is the slope of the line and b is the y-intercept. The slope of a line is a measure of how steep the line is, and it is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.
Finding the Slope of the Given Line
To find the slope of the given line, we need to rewrite the equation in the slope-intercept form. The given equation is y - 19/6 = 2(x + 7/2). To rewrite this equation, we need to isolate y on one side of the equation.
y - \frac{19}{6} = 2\left(x + \frac{7}{2}\right)
First, we need to distribute the 2 to the terms inside the parentheses:
y - \frac{19}{6} = 2x + \frac{14}{2}
Next, we need to simplify the right-hand side of the equation by combining the terms:
y - \frac{19}{6} = 2x + 7
Now, we need to add 19/6 to both sides of the equation to isolate y:
y = 2x + 7 + \frac{19}{6}
To add 7 and 19/6, we need to find a common denominator, which is 6. So, we can rewrite 7 as 42/6:
y = 2x + \frac{42}{6} + \frac{19}{6}
Now, we can combine the terms on the right-hand side of the equation:
y = 2x + \frac{61}{6}
Finding the Slope of the Parallel Line
Since the line we are looking for is parallel to the given line, it has the same slope as the given line. The slope of the given line is 2, so the slope of the parallel line is also 2.
Finding the Equation of the Parallel Line
Now that we have the slope of the parallel line, we can use the point-slope form of a line to find its equation. 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.
We are given that the line passes through the point (4, -1), so we can substitute x1 = 4 and y1 = -1 into the equation:
y - (-1) = 2(x - 4)
Simplifying the equation, we get:
y + 1 = 2(x - 4)
Now, we need to distribute the 2 to the terms inside the parentheses:
y + 1 = 2x - 8
Next, we need to add 8 to both sides of the equation to isolate y:
y = 2x - 8 + 1
Simplifying the equation, we get:
y = 2x - 7
Conclusion
In this article, we found the slope-intercept form of the equation of a line that passes through a given point and is parallel to another line. We first found the slope of the given line by rewriting its equation in the slope-intercept form. Then, we used the point-slope form of a line to find the equation of the parallel line. The final equation of the parallel line is y = 2x - 7.
Key Takeaways
- The slope-intercept form of a line is given by the equation y = mx + b, where m is the slope of the line and b is the y-intercept.
- Parallel lines have the same slope but different y-intercepts.
- 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.
- To find the equation of a line that passes through a given point and is parallel to another line, we need to find the slope of the given line and use the point-slope form of a line to find the equation of the parallel line.
Further Reading
- For more information on the slope-intercept form of a line, see Slope-Intercept Form.
- For more information on parallel lines, see Parallel Lines.
- For more information on the point-slope form of a line, see Point-Slope Form.
Q: What is the slope-intercept form of a line?
A: The slope-intercept form of a line is given by the equation y = mx + b, where m is the slope of the line and b is the y-intercept.
Q: What is the slope of a line?
A: The slope of a line is a measure of how steep the line is, and it is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.
Q: How do I find the slope of a line?
A: To find the slope of a line, you need to rewrite the equation of the line in the slope-intercept form. This involves isolating y on one side of the equation and then identifying the slope (m) and the y-intercept (b).
Q: What is the point-slope form of a line?
A: 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.
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 find the slope of the given line and use the point-slope form of a line to find the equation of the parallel line.
Q: What is the difference between a slope and a y-intercept?
A: The slope of a line is a measure of how steep the line is, while the y-intercept is the point where the line intersects the y-axis.
Q: How do I determine if two lines are parallel?
A: Two lines are parallel if they have the same slope but different y-intercepts.
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 does not change in the vertical direction.
Q: Can a line have an undefined slope?
A: Yes, a line can have an undefined slope. This means that the line is vertical and does not change in the horizontal direction.
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 need to use the point-slope form of a line and substitute the given point and slope into the equation.
Q: What is the relationship between the slope of a line and its graph?
A: The slope of a line is a measure of how steep the line is, and it is related to the graph of the line. A line with a positive slope will rise from left to right, while a line with a negative slope will fall from left to right.
Q: Can a line have a slope of one?
A: Yes, a line can have a slope of one. This means that the line is a 45-degree angle and changes equally in the vertical and horizontal directions.
Q: Can a line have a slope of -1?
A: Yes, a line can have a slope of -1. This means that the line is a 45-degree angle and changes equally in the vertical and horizontal directions, but in the opposite direction.
Q: How do I find the equation of a line that passes through two given points?
A: To find the equation of a line that passes through two given points, you need to find the slope of the line using the two points and then use the point-slope form of a line to find the equation of the line.
Q: What is the relationship between the slope of a line and its y-intercept?
A: The slope of a line is a measure of how steep the line is, while the y-intercept is the point where the line intersects the y-axis. The slope and y-intercept are related through the equation y = mx + b, where m is the slope and b is the y-intercept.
Q: Can a line have a y-intercept of zero?
A: Yes, a line can have a y-intercept of zero. This means that the line passes through the origin (0, 0) and does not intersect the y-axis.
Q: Can a line have a y-intercept of a non-zero value?
A: Yes, a line can have a y-intercept of a non-zero value. This means that the line intersects the y-axis at a point other than the origin (0, 0).
Q: How do I determine if a line is horizontal or vertical?
A: A line is horizontal if it has a slope of zero and does not change in the vertical direction. A line is vertical if it has an undefined slope and does not change in the horizontal direction.
Q: Can a line be both horizontal and vertical?
A: No, a line cannot be both horizontal and vertical at the same time. A line can be either horizontal or vertical, but not both.
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 find the slope of the given line and then use the point-slope form of a line to find the equation of the perpendicular line.
Q: What is the relationship between the slope of a line and its perpendicular line?
A: The slope of a line is the negative reciprocal of the slope of its perpendicular line. This means that if the slope of a line is m, then the slope of its perpendicular line is -1/m.
Q: Can a line have a perpendicular line with a slope of zero?
A: No, a line cannot have a perpendicular line with a slope of zero. This is because a line with a slope of zero is horizontal, and a horizontal line does not have a perpendicular line.
Q: Can a line have a perpendicular line with an undefined slope?
A: No, a line cannot have a perpendicular line with an undefined slope. This is because a line with an undefined slope is vertical, and a vertical line does not have a perpendicular line.
Q: How do I determine if a line is parallel or perpendicular to another line?
A: Two lines are parallel if they have the same slope but different y-intercepts. Two lines are perpendicular if the slope of one line is the negative reciprocal of the slope of the other line.
Q: Can a line be both parallel and perpendicular to another line?
A: No, a line cannot be both parallel and perpendicular to another line at the same time. A line can be either parallel or perpendicular to another line, but not both.
Q: How do I find the equation of a line that passes through a given point and is parallel or perpendicular to another line?
A: To find the equation of a line that passes through a given point and is parallel or perpendicular to another line, you need to find the slope of the given line and then use the point-slope form of a line to find the equation of the parallel or perpendicular line.
Q: What is the relationship between the slope of a line and its parallel or perpendicular line?
A: The slope of a line is the same as the slope of its parallel line, and it is the negative reciprocal of the slope of its perpendicular line.
Q: Can a line have a parallel line with a slope of zero?
A: No, a line cannot have a parallel line with a slope of zero. This is because a line with a slope of zero is horizontal, and a horizontal line does not have a parallel line.
Q: Can a line have a perpendicular line with a slope of zero?
A: No, a line cannot have a perpendicular line with a slope of zero. This is because a line with a slope of zero is horizontal, and a horizontal line does not have a perpendicular line.
Q: Can a line have a parallel line with an undefined slope?
A: No, a line cannot have a parallel line with an undefined slope. This is because a line with an undefined slope is vertical, and a vertical line does not have a parallel line.
Q: Can a line have a perpendicular line with an undefined slope?
A: No, a line cannot have a perpendicular line with an undefined slope. This is because a line with an undefined slope is vertical, and a vertical line does not have a perpendicular line.
Q: How do I determine if a line is a function or not?
A: A line is a function if it passes the vertical line test, which means that for every x-value, there is only one corresponding y-value.
Q: Can a line be a function and not pass the vertical line test?
A: No, a line cannot be a function and not pass the vertical line test. If a line does not pass the vertical line test, it is not a function.
Q: How do I find the equation of a line that passes through a given point and is a function?
A: To find the equation of a line that passes through a given point and is a function, you need to find the slope of the line and then use the point-slope form of a line to find the equation of the line.