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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