A line segment is bisected by a line with the equation # - 3 y + 6 x = 6 #. If one end of the line segment is at #( 3 , 3 )#, where is the other end?

Answer 1

The other end could be any point on the line #y=2x-1#

For convenience, I will rearrange the given equation
#color(white)("XXX")-3y+6x=6#
as
#color(white)("XXX")y=2x-2#

Consider the vertical line through #(3,3)#.
Since #x# is constant for all points on a vertical line,
this vertical line will intersect #y=2x-2# at #(3,4)#

The distance between #(3,3)# and #(3,4)# is #1# unit.

The point #(3,5)# is also on this vertical line at a distance of #2# units from #(3,3)#.

Therefore #y=2x-2# (originally given as #-3y+6x=6#) bisects the line segment joining #(3,3)# and #(3,5)#.

Therefore one possible endpoint would be at #(3,5)#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Perhaps less obviously, any point on a line through #(3,5)# and parallel to the given line #y=2x-2# (or in its original but less convenient form #-3y+6x=6#)
will also be a line segment endpoint bisected by the given equation.

Which can be simplified as
#color(white)("XXX")y=2x-1#

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Answer 2

To find the other end of the line segment, you can solve the given equation for y to obtain the equation of the line segment. Then, substitute the x-coordinate of the given point into the equation to find the corresponding y-coordinate.

First, solve the equation -3y + 6x = 6 for y: -3y = -6x + 6 y = 2x - 2

Next, substitute the x-coordinate of the given point (3, 3) into the equation: y = 2(3) - 2 y = 6 - 2 y = 4

So, the other end of the line segment is at the point (3, 4).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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