Name: Sophia RamirezWrite An Equation Of The Line That Passes Through The Given Point And Is Parallel To The Given Line. Use Slope-intercept Form.1. Given Point: \[$(-2, 5)\$\], Line: \[$2y = 4x - 6\$\]2. Given Point: \[$(9,
Introduction
In mathematics, the concept of parallel lines is crucial in understanding various geometric and algebraic relationships. When given a point and a line, we can determine the equation of a line that passes through the given point and is parallel to the given line. In this article, we will explore how to find the equation of a line in slope-intercept form that is parallel to a given line and passes through a given point.
Understanding Slope-Intercept Form
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 represents the rate of change of the line, and the y-intercept represents the point where the line intersects the y-axis.
Finding the Slope of a Line
To find the slope of a line, we can use the slope formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are two points on the line.
Example 1: Finding the Equation of a Parallel Line
Given point: (-2, 5) Line: 2y = 4x - 6
First, we need to find the slope of the given line. To do this, we can rewrite the equation in slope-intercept form:
2y = 4x - 6 y = 2x - 3
Now, we can see that the slope of the line is 2. Since the line we want to find is parallel to the given line, it will have the same slope.
Next, we need to find the equation of the line that passes through the given point (-2, 5) and has a slope of 2. We can use the point-slope form of a line:
y - y1 = m(x - x1)
where (x1, y1) is the given point and m is the slope.
y - 5 = 2(x - (-2)) y - 5 = 2(x + 2) y - 5 = 2x + 4
Now, we can rewrite the equation in slope-intercept form:
y = 2x + 9
Therefore, the equation of the line that passes through the given point (-2, 5) and is parallel to the given line is y = 2x + 9.
Example 2: Finding the Equation of a Parallel Line
Given point: (9, 2) Line: y = 3x - 1
First, we need to find the slope of the given line. To do this, we can rewrite the equation in slope-intercept form:
y = 3x - 1
Now, we can see that the slope of the line is 3. Since the line we want to find is parallel to the given line, it will have the same slope.
Next, we need to find the equation of the line that passes through the given point (9, 2) and has a slope of 3. We can use the point-slope form of a line:
y - y1 = m(x - x1)
where (x1, y1) is the given point and m is the slope.
y - 2 = 3(x - 9) y - 2 = 3(x - 9) y - 2 = 3x - 27 y = 3x - 25
Therefore, the equation of the line that passes through the given point (9, 2) and is parallel to the given line is y = 3x - 25.
Conclusion
In conclusion, finding the equation of a line that passes through a given point and is parallel to a given line involves finding the slope of the given line and using the point-slope form of a line to find the equation of the line that passes through the given point and has the same slope. By following these steps, we can determine the equation of a line in slope-intercept form that is parallel to a given line and passes through a given point.
Key Takeaways
- The slope-intercept form of a line is given by the equation y = mx + b, where m is the slope and b is the y-intercept.
- The slope of a line represents the rate of change of the line, and the y-intercept represents the point where the line intersects the y-axis.
- To find the equation of a line that passes through a given point and is parallel to a given 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 line that passes through the given point and has the same slope.
- The equation of a line that passes through a given point and is parallel to a given line can be found using the point-slope form of a line: y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope.
Final Thoughts
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 and b is the y-intercept.
Q: How do I find the slope of a line?
A: To find the slope of a line, you can use the slope 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 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 a given line?
A: To find the equation of a line that passes through a given point and is parallel to a given 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 line that passes through the given point and has the same slope.
Q: What is the difference between a parallel line and a perpendicular line?
A: A parallel line is a line that has the same slope as another line, but does not intersect it. A perpendicular line is a line that has a slope that is the negative reciprocal of another line, and intersects it at a 90-degree angle.
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 line: y - y1 = m(x - x1), where (x1, y1) and (x2, y2) are the two points.
Q: What is the y-intercept of a line?
A: The y-intercept of a line is the point where the line intersects the y-axis. It is represented by the value of b in the slope-intercept form of a line: y = mx + b.
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 line: y - y1 = m(x - x1), where (x1, y1) is the given point and m is the given slope.
Q: What is the significance of the slope of a line?
A: The slope of a line represents the rate of change of the line. It tells us how much the y-coordinate changes when the x-coordinate changes by a certain amount.
Q: How do I determine if two lines are parallel or perpendicular?
A: To determine if two lines are parallel or perpendicular, you can compare their slopes. If the slopes are equal, the lines are parallel. If the slopes are negative reciprocals of each other, the lines are perpendicular.
Q: What is the equation of a line that passes through the origin and has a given slope?
A: The equation of a line that passes through the origin and has a given slope is given by the equation y = mx, where m is the slope.
Q: How do I find the equation of a line that passes through a given point and is perpendicular to a given line?
A: To find the equation of a line that passes through a given point and is perpendicular to a given line, you need to find the slope of the given line and use the negative reciprocal of the slope to find the equation of the line that passes through the given point and is perpendicular to the given line.
Q: What is the significance of the y-intercept of a line?
A: The y-intercept of a line represents the point where the line intersects the y-axis. It is a crucial point in understanding the behavior of the line.
Q: How do I determine if a line is horizontal or vertical?
A: To determine if a line is horizontal or vertical, you can look at its slope. If the slope is 0, the line is horizontal. If the slope is undefined, the line is vertical.
Q: What is the equation of a horizontal line?
A: The equation of a horizontal line is given by the equation y = b, where b is the y-intercept.
Q: What is the equation of a vertical line?
A: The equation of a vertical line is given by the equation x = a, where a is the x-intercept.
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 line: y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope. Then, you can substitute the value of the y-intercept into the equation to find the equation of the line.