# A line segment is bisected by a line with the equation # - 6 y + 2 x = 3 #. If one end of the line segment is at #( 4 , 8 )#, where is the other end?

Any point on the line

A line drawn through a point that has the potential to be an endpoint and runs parallel to the bisecting line will yield all possible endpoints for the bisected line segment.

The point and the two lines in question are shown in the following graph: graph{(-6y+2x-3)(3y-x+27)((x-4)^2+(y-8)^2-0.02)=0 [-25.65, 25.64, -12.83, 12.81]}

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To find the other end of the line segment bisected by the line with the equation ( -6y + 2x = 3 ), given that one end is at the point (4, 8), you can use the midpoint formula.

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 (4, 8), and knowing that the midpoint lies on the line ( -6y + 2x = 3 ), we can find the coordinates of the other end by finding the equation of the line passing through the given point and perpendicular to the given line. Then, we can find the intersection point of this perpendicular line with the given line.

First, we find the slope of the given line: ( m = \frac{1}{3} ). The slope of the line perpendicular to this is ( -3 ) (negative reciprocal).

Using the point-slope form, we find the equation of the line passing through (4, 8) with slope ( -3 ): [ y - 8 = -3(x - 4) ]

Simplify this to get: [ y = -3x + 20 ]

Now, we need to solve the system of equations: [ \begin{cases} -6y + 2x = 3 \ y = -3x + 20 \end{cases} ]

Solving this system will give us the coordinates of the other end of the line segment.

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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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