A line segment is bisected by a line with the equation #  3 y + 5 x = 8 #. If one end of the line segment is at #( 7 , 9 )#, where is the other end?
Coordinates of the other end point
The bisecting line is assumed to be a perpendicular bisector.
Equations (1) and (2) solved,
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To find the other end of the line segment bisected by the line ( 3y + 5x = 8 ), knowing that one end is at (7, 9), follow these steps:
 Find the slope of the line ( 3y + 5x = 8 ) by rearranging it into slopeintercept form ( y = mx + b ), where ( m ) is the slope.
 Use the fact that the line bisects the line segment to find the midpoint of the line segment.
 Use the midpoint formula to find the coordinates of the other end of the line segment.
Let's proceed with these steps:

Rearrange the equation ( 3y + 5x = 8 ) into slopeintercept form: [ 3y = 5x + 8 ] [ y = \frac{5}{3}x  \frac{8}{3} ]
The slope of the line is ( m = \frac{5}{3} ).

Since the line bisects the line segment, its slope is perpendicular to the segment. Thus, the negative reciprocal of ( \frac{5}{3} ) is the slope of the line segment.
The negative reciprocal of ( \frac{5}{3} ) is ( \frac{3}{5} ).

Use the midpoint formula to find the other end of the line segment: [ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) ]
Given one end at (7, 9) and the midpoint, we can find the other end.
Let ( (x, y) ) be the coordinates of the other end. [ \text{Midpoint} = \left( \frac{7 + x}{2}, \frac{9 + y}{2} \right) ]
Since the line bisects the line segment, its midpoint must lie on the line ( 3y + 5x = 8 ). Plug in the coordinates of the midpoint into the equation of the line and solve for ( x ) and ( y ).
[ 3\left(\frac{9 + y}{2}\right) + 5\left(\frac{7 + x}{2}\right) = 8 ]
Simplify and solve for ( x ) and ( y ).
After finding the values of ( x ) and ( y ), you will have the coordinates of the other end of the line segment.
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