Write the equation of a line that is perpendicular to the given line and that passes through the given point.
y-4=5/2(x+3);(-7,8)
I need help 🙂
the given line has slope 5/2
Any line perpendicular to that has slope -2/5
So, now you have a point and a slop -- use the point-slope form for the desired line:
y-8 = -2/5 (x+7)
Alright, let's tackle this problem while having some fun! 🎉
First, we need to find the slope of the given line. The equation is in the form y = mx + b, where m represents the slope. In this case, we have y - 4 = (5/2)(x + 3). To convert it to slope-intercept form, we isolate the y-term:
y - 4 = (5/2)(x + 3)
y - 4 = (5/2)x + (15/2)
y = (5/2)x + (15/2) + 4
y = (5/2)x + (15/2) + 8/2
y = (5/2)x + (23/2)
So, the slope of the given line is 5/2. To find the slope of a line perpendicular to it, we'll take the negative reciprocal of 5/2. That means the perpendicular slope is -2/5.
Now, let's use the point (-7, 8) to find the equation of the perpendicular line. We'll use the point-slope form, y - y1 = m(x - x1), where (x1, y1) is the given point and m is the perpendicular slope.
Let's plug in the values:
y - 8 = -2/5(x - (-7))
y - 8 = -2/5(x + 7)
y - 8 = -2/5x - 14/5
To get rid of the fraction, let's multiply the entire equation by 5:
5(y - 8) = -2x - 14
5y - 40 = -2x - 14
5y = -2x - 14 + 40
5y = -2x + 26
Finally, let's rearrange it to slope-intercept form, y = mx + b:
5y = -2x + 26
y = (-2/5)x + 26/5
And there you have it! The equation of the line perpendicular to the original line, passing through the point (-7, 8), is y = (-2/5)x + 26/5. Hope that brought a smile to your face! 🤡
To find the equation of a line that is perpendicular to the given line y-4=5/2(x+3), we need to determine the slope of the given line.
The given line is in slope-intercept form (y = mx + b), where m represents the slope. So, let's rewrite the given line in slope-intercept form:
y - 4 = 5/2(x + 3)
y - 4 = (5/2)x + (5/2)(3)
y - 4 = (5/2)x + 15/2
y = (5/2)x + 15/2 + 4
y = (5/2)x + 15/2 + 8/2
y = (5/2)x + 23/2
From this equation, we can determine that the slope of the given line is 5/2.
Since we are looking for a line that is perpendicular to this line, we know that the slope of the perpendicular line will be the negative reciprocal of the slope of the given line.
The negative reciprocal of 5/2 is -2/5.
Now, we have the slope (-2/5) of the perpendicular line and a point (-7, 8) that it passes through. We can use the point-slope form to find the equation of the perpendicular line.
The point-slope form is given by y - y1 = m(x - x1), where (x1, y1) represents the given point and m is the slope.
Plugging in the values, we have:
y - 8 = (-2/5)(x - (-7))
y - 8 = (-2/5)(x + 7)
y - 8 = (-2/5)x - (2/5)(7)
y - 8 = (-2/5)x - 14/5
y = (-2/5)x - 14/5 + 40/5
y = (-2/5)x + 26/5
Therefore, the equation of the line that is perpendicular to y - 4 = 5/2(x + 3) and passes through the point (-7, 8) is y = (-2/5)x + 26/5.
To find the equation of a line that is perpendicular to the given line and passes through the given point, you need to follow these steps:
Step 1: Determine the slope of the given line.
The given line equation is in the slope-intercept form (y = mx + b), where 'm' represents the slope. By comparing the equation with the slope-intercept form, you can determine that the slope of the given line is 5/2.
Step 2: Determine the slope of the line perpendicular to the given line.
The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. So, to find the slope of the line perpendicular to the given line, take the negative reciprocal of 5/2.
The negative reciprocal of 5/2 can be found by flipping the fraction and changing its sign:
Negative reciprocal = -2/5.
Step 3: Use the slope and the given point to determine the equation of the perpendicular line.
You now have the slope (-2/5) and the given point (-7, 8). To find the equation of the line, use the point-slope form (y - y1 = m(x - x1)), where (x1, y1) is the given point and 'm' is the slope.
Plugging in the values:
y - 8 = (-2/5)(x - (-7)).
Simplifying further:
y - 8 = (-2/5)(x + 7).
Expanding the equation:
y - 8 = (-2/5)x - (2/5)(7).
Simplifying:
y - 8 = (-2/5)x - 14/5.
Adding 8 to both sides:
y = (-2/5)x - 14/5 + 8.
Simplifying further:
y = (-2/5)x - 14/5 + 40/5.
Combining the terms on the right side:
y = (-2/5)x + 26/5.
So, the equation of the line that is perpendicular to the given line and passes through the point (-7, 8) is:
y = (-2/5)x + 26/5.