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?

1 Answer
Dean R.
May 11, 2018

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

Explanation:

#-7 y + 5x = 1 #

I find it less confusing this way:

#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 #

We find the meet by multiplying the first by 5 and the second by 7:

#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#

If we call our endpoint E and our meet M we get an informal equation for the other endpoint F that's

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

So our other endpoint is

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

Check:

Let's see if we can get the grapher to graph it:

#-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 y + 5x - 1) ( (y-4)(157/37 -1) - (x-1)( -20/37 -4 ) ) = 0 [-7.83, 12.17, -2.44, 7.56]}

Looks pretty good.

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