Use The Given Conditions To Write An Equation For The Line In Point-slope Form And In Slope-intercept Form.The Line Passes Through { (9, -8)$}$ And Is Perpendicular To The Line Whose Equation Is { Y = \frac{1}{5}x + 3$}$.1. Write An
Introduction
In this article, we will explore the concept of finding the equation of a line that is perpendicular to a given line. We will use the point-slope form and slope-intercept form to write the equation of the perpendicular line. The given line is in slope-intercept form, and we will use this information to find the slope of the perpendicular line.
Step 1: Find the Slope of the Given Line
The given line is in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. In this case, the equation of the given line is y = (1/5)x + 3. Therefore, the slope of the given line is 1/5.
Step 2: Find the Slope of the Perpendicular Line
The slope of the perpendicular line is the negative reciprocal of the slope of the given line. To find the negative reciprocal, we take the reciprocal of the slope and change its sign. Therefore, the slope of the perpendicular line is -5.
Step 3: Write the Equation of the Perpendicular Line in Point-Slope Form
The point-slope form of a line is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. We are given a point on the line, which is (9, -8). We will use this point and the slope of the perpendicular line to write the equation of the line in point-slope form.
y - (-8) = -5(x - 9)
Step 4: Simplify the Equation
To simplify the equation, we will distribute the slope to the terms inside the parentheses.
y + 8 = -5x + 45
Step 5: Write the Equation in Slope-Intercept Form
To write the equation in slope-intercept form, we will isolate the y-term on one side of the equation.
y = -5x + 45 - 8
y = -5x + 37
Conclusion
In this article, we have found the equation of a line that is perpendicular to a given line. We used the point-slope form and slope-intercept form to write the equation of the perpendicular line. The slope of the perpendicular line is the negative reciprocal of the slope of the given line. We have also simplified the equation and written it in slope-intercept form.
Key Takeaways
- The slope of the perpendicular line is the negative reciprocal of the slope of the given line.
- The point-slope form of a line is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
- The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept.
Example Problems
- Find the equation of a line that is perpendicular to the line whose equation is y = 2x + 1. The line passes through the point (3, 5).
- Find the equation of a line that is perpendicular to the line whose equation is y = -3x + 2. The line passes through the point (4, 6).
Solutions
- The slope of the given line is 2. The slope of the perpendicular line is -1/2. The equation of the perpendicular line in point-slope form is y - 5 = (-1/2)(x - 3). Simplifying the equation, we get y = (-1/2)x + 11/2.
- The slope of the given line is -3. The slope of the perpendicular line is 1/3. The equation of the perpendicular line in point-slope form is y - 6 = (1/3)(x - 4). Simplifying the equation, we get y = (1/3)x + 14/3.
Frequently Asked Questions (FAQs) =====================================
Q: What is the point-slope form of a line?
A: The point-slope form of a line is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
Q: What is the slope-intercept form of a line?
A: The slope-intercept form of a line is 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 formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.
Q: How do I find the equation of a line that is perpendicular to a given line?
A: To find the equation of a line that is perpendicular to a given line, you need to find the slope of the given line and then take the negative reciprocal of that slope. You can then use the point-slope form or slope-intercept form to write the equation of the perpendicular line.
Q: What is the negative reciprocal of a slope?
A: The negative reciprocal of a slope is the reciprocal of the slope with a negative sign. For example, if the slope is 2, the negative reciprocal is -1/2.
Q: How do I simplify an equation in point-slope form?
A: To simplify an equation in point-slope form, you need to distribute the slope to the terms inside the parentheses and then combine like terms.
Q: How do I write an equation in slope-intercept form?
A: To write an equation in slope-intercept form, you need to isolate the y-term on one side of the equation and then combine like terms.
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 find the y-intercept of a line?
A: To find the y-intercept of a line, you can set x = 0 in the equation of the line and solve for y.
Q: Can a line have a slope of 0?
A: Yes, a line can have a slope of 0. This means that the line is horizontal and does not change in the y-direction.
Q: Can a line have a slope of infinity?
A: No, a line cannot have a slope of infinity. However, a line can have a vertical slope, which means that it is a vertical line that does not change in the x-direction.
Q: How do I graph a line?
A: To graph a line, you need to plot two points on the line and then draw a line through those points. You can also use the slope-intercept form to graph a line by plotting the y-intercept and then using the slope to determine the direction of the line.
Q: What is the equation of a line that is parallel to a given line?
A: The equation of a line that is parallel to a given line is the same as the equation of the given line, except for the y-intercept. The slope of the parallel line is the same as the slope of the given line.
Q: What is the equation of a line that is perpendicular to a given line?
A: The equation of a line that is perpendicular to a given line is the negative reciprocal of the slope of the given line. You can then use the point-slope form or slope-intercept form to write the equation of the perpendicular line.