A line segment is bisected by a line with the equation # -7 y + 5 x = 1 #. If one end of the line segment is at #(1 ,4 )#, where is the other end?

Question

A line segment is bisected by a line with the equation #     -7 y  +  5  x   =   1  #.  If one end of the line segment is at #(1  ,4  )#, where is the other end?

Jayden Jackson · Accepted Answer

I get #(157/37 , -20/37 ) #

#-7 y + 5x = 1 #

This way, it makes more sense to me:

#5x - 7y = 1 #

The perpendicular family is gotten by swapping the coefficients on #x# and #y#, negating one. The constant is gotten by plugging in the point #(1,4)# on the perpendicular:

#7x + 5y = 7(1) + 5(4) = 27 #

By multiplying the first by 5 and the second by 7, we can determine the meet:

#25 x - 35 y = 5 #

#49 x + 35y = 7 cdot 27 = 189#

Adding,

#74 x = 194 #

#x = 194/74 = 97/37#

#y = 1/5 (27 - 7(97/37)) = 64/37#

An approximate equation for the other endpoint F can be obtained by calling our endpoint E and our meet M.

# F = M-(E-M)= M + (M-E) = 2M - E #

Thus, the other endpoint we have is

# (2 (97/37) - 1, 2( 64/37) -4 ) = (157/37 , -20/37 ) #

Check:

Let's attempt to graph it using the grapher:

#-7 y + 5x = 1 #

# (y-4)(157/37 -1)=(x-1)( -20/37 -4 )#

# ( -7 y + 5x - 1) ( (y-4)(157/37 -1) - (x-1)( -20/37 -4 ) ) = 0 #

graph{(-7.83, 12.17, -2.44, 7.56]} = (x-1)( -20/37 -4 ) - (y-4)(157/37 -1).

It appears quite nice.

Autumn Armstrong · Answer

undefined To find the other end of the line segment bisected by the line -7y + 5x = 1, we first need to find the equation of the perpendicular bisector of the line segment. This perpendicular bisector will pass through the midpoint of the line segment.

To find the midpoint, we use the midpoint formula:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)

Given one end of the line segment is (1, 4), let's call it point A. The coordinates of point A are (x1, y1) = (1, 4).

Let's denote the other end of the line segment as point B, with coordinates (x, y).

Now, we'll find the midpoint of the line segment using the midpoint formula:

Midpoint = ((1 + x)/2, (4 + y)/2)

The coordinates of the midpoint are the averages of the x-coordinates and y-coordinates of the endpoints of the line segment.

Next, we need to find the slope of the line -7y + 5x = 1. The equation is given in the form Ax + By = C, where A = 5, B = -7, and C = 1. We rearrange the equation into slope-intercept form (y = mx + b) to find the slope (m).

Slope (m) = -A/B

Substituting the values, we get:
Slope (m) = -5/-7 = 5/7

The perpendicular bisector of the line segment will have a slope that is the negative reciprocal of the slope of the given line. So, the slope of the perpendicular bisector is -7/5.

Now, we have the midpoint and the slope of the perpendicular bisector. We can use these to find the equation of the perpendicular bisector. Then, we can solve the system of equations formed by the equation of the perpendicular bisector and the given line to find the coordinates of point B.