Find an equation of the line (in slope-intercept form) that passes through the point (7,5) and is perpendicular to the line 7x + 10y - 17 = 0
y=?
consider the line:
7x+10y-17=0
y=-.7x+1.7
so the slope of the perpendicular line is
m=+10/7
y=10/7 x + b put in y, x given, solve for b, and you have the equation of the lline
y=10x/7 + b
To find the equation of the line that is perpendicular to the line 7x + 10y - 17 = 0, we need to find the slope of the given line.
First, let's rearrange the equation into slope-intercept form, which is y = mx + b.
7x + 10y - 17 = 0
10y = -7x + 17
y = (-7/10)x + 17/10
The slope of this line is -7/10.
Since the line we want is perpendicular to this line, the slope of the new line will be the negative reciprocal of -7/10.
Let's call the new slope m_perpendicular.
m_perpendicular = -1 / (-7/10)
m_perpendicular = 10/7
So, the slope of the new line is 10/7.
Now we have the slope (m_perpendicular) and a point (7,5) that the line passes through. We can use the point-slope form of the equation, which is y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope.
Using the point (7,5) and the slope 10/7 in the point-slope form, we get:
y - 5 = (10/7)(x - 7)
Now, we can rearrange this equation into slope-intercept form to find y:
y - 5 = (10/7)x - 10
y = (10/7)x - 10 + 5
y = (10/7)x - 5
Therefore, the equation of the line that passes through the point (7,5) and is perpendicular to the line 7x + 10y - 17 = 0 is y = (10/7)x - 5.
To find the equation of a line that is perpendicular to a given line, we need to determine the slope of the given line and then find the negative reciprocal of that slope.
First, let's rearrange the given equation into slope-intercept form (y = mx + b), where m represents the slope of the line:
7x + 10y - 17 = 0
To isolate y, we'll subtract 7x from both sides:
10y = -7x + 17
Dividing through by 10:
y = (-7/10)x + 17/10
Thus, the slope of the given line is -7/10.
The line we want to find is perpendicular to this line, so its slope will be the negative reciprocal of -7/10. To find the negative reciprocal, we flip the fraction and change the sign:
m_perpendicular = -1 / (-7/10)
Now, simplify by multiplying the numerator and denominator by -10/7:
m_perpendicular = 10/7
We now have the slope of the perpendicular line.
To find the equation of the line that goes through the point (7,5) and has a slope of 10/7, we can use the point-slope form of a line:
y - y1 = m(x - x1)
Substituting the coordinates of the point (7,5):
y - 5 = (10/7)(x - 7)
Expanding and rearranging:
y - 5 = (10/7)x - (10/7)(7)
y - 5 = (10/7)x - 10
Adding 5 to both sides:
y = (10/7)x - 10 + 5
Simplifying:
y = (10/7)x - 5
Therefore, the equation of the line that passes through the point (7,5) and is perpendicular to the line 7x + 10y - 17 = 0 is y = (10/7)x - 5.