# A line segment is bisected by a line with the equation # - 5 y + 3 x = 1 #. If one end of the line segment is at #(6 ,4 )#, where is the other end?

If

if

Then for any point

the

will be the same as

Since

for any point

the corresponding target end point will be

Specifically we can see that

with a corresponding target point of

Note also that the target line is parallel to

and therefore has the same slope (namely

Using the slope-point form for the target line, we get

or

or

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To find the other end of the line segment bisected by the line ( -5y + 3x = 1 ) given that one end is at (6, 4), we can first find the equation of the line segment connecting the given point to the point of intersection with the line ( -5y + 3x = 1 ). Then, using the midpoint formula, we can find the other end of the line segment.

First, find the point of intersection of the given line ( -5y + 3x = 1 ) and the line segment. To do this, solve the system of equations formed by ( -5y + 3x = 1 ) and the line connecting the given point (6, 4).

Substitute ( y = \frac{3x - 1}{5} ) into the equation of the line segment and solve for ( x ):

[ 4 = \frac{3x - 1}{5} ] [ 20 = 3x - 1 ] [ 21 = 3x ] [ x = 7 ]

Now, substitute ( x = 7 ) into the equation of the line segment to find ( y ):

[ -5y + 3(7) = 1 ] [ -5y + 21 = 1 ] [ -5y = -20 ] [ y = 4 ]

So, the point of intersection is (7, 4).

Now, use the midpoint formula to find the other end of the line segment. The midpoint formula is:

[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) ]

Given one end of the line segment as (6, 4) and the point of intersection as (7, 4), the other end can be found as:

[ \text{Midpoint} = \left( \frac{6 + 7}{2}, \frac{4 + 4}{2} \right) ] [ \text{Midpoint} = \left( \frac{13}{2}, \frac{8}{2} \right) ] [ \text{Midpoint} = \left( \frac{13}{2}, 4 \right) ]

So, the other end of the line segment is at ( \left( \frac{13}{2}, 4 \right) ).

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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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