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

Exact value:

Decimal value (-2.16 9.77)

Currently, locate the other endpoint:

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To find the other end of the line segment bisected by the line with the equation (4y - 6x = 8) given one end at (8, 3), follow these steps:

- Calculate the slope of the line given by the equation (4y - 6x = 8).
- Determine the midpoint of the line segment using the given point and the slope of the line.
- Use the midpoint and the given point to find the other end of the line segment.

First, rearrange the equation (4y - 6x = 8) into slope-intercept form (y = mx + b), where (m) is the slope and (b) is the y-intercept.

[4y - 6x = 8] [4y = 6x + 8] [y = \frac{3}{2}x + 2]

The slope of the line is (\frac{3}{2}).

The midpoint formula for a line segment with endpoints ((x_1, y_1)) and ((x_2, y_2)) is given by:

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

Given one end at (8, 3), we can use the midpoint formula to find the midpoint.

[ \text{Midpoint} = \left( \frac{8 + x_2}{2}, \frac{3 + y_2}{2} \right) ]

Since the line bisects the segment, the slope of the line from (8, 3) to the midpoint is the negative reciprocal of (\frac{3}{2}), which is (-\frac{2}{3}).

Now, using the point-slope form of a line ((y - y_1 = m(x - x_1))), we can find the equation of the line passing through the given point (8, 3) with the slope (-\frac{2}{3}).

[y - 3 = -\frac{2}{3}(x - 8)]

[y - 3 = -\frac{2}{3}x + \frac{16}{3}]

[y = -\frac{2}{3}x + \frac{16}{3} + 3]

[y = -\frac{2}{3}x + \frac{25}{3}]

Next, set this equation equal to the equation of the line given in the question to find the intersection point.

[ -\frac{2}{3}x + \frac{25}{3} = \frac{3}{2}x + 2 ]

Solve for (x), then substitute the value of (x) back into either equation to find (y). This will give 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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