How do you find the equation of the line that passes through the point (1, -5). and is perpendicular to the line #y=-1/2x+4#?

Question

Nathan Axel · Accepted Answer

The required equation is #y=2x-7#

Given -

#y=-1/2x+4#

Slope of the given line #m_1=-1/2#

Slope of the perpendicular line #m_2=-1 xx 1/(m_1)#

#m_2=-1xx (-2/1)=2#

Then -

Let the equation of the required line be -

#y=mx+c#

Given the point #(1, -5)# we have to find the value of #c#

#2 xx 1+c=-5# #c=-5-2=-7#

The required equation is #y=2x-7#

Nathan Axel · Answer

undefined To find the equation of a line perpendicular to a given line, you need to determine the negative reciprocal of the slope of the given line. The given line has a slope of -1/2, so the perpendicular line will have a slope that is the negative reciprocal, which is 2.

Then, using the point-slope form of the equation of a line, which is $ y - y_1 = m(x - x_1) $, where $ m $ is the slope and $ (x_1, y_1) $ is a point on the line, substitute the given point (1, -5) and the perpendicular slope (2) into the equation:

$$ y - (-5) = 2(x - 1) $$

$$ y + 5 = 2(x - 1) $$

To convert this equation into slope-intercept form (y = mx + b), distribute 2:

$$ y + 5 = 2x - 2 $$

$$ y = 2x - 7 $$

So, the equation of the line that passes through the point (1, -5) and is perpendicular to the line $ y = -\frac{1}{2}x + 4 $ is $ y = 2x - 7 $.