A line segment is bisected by a line with the equation # - y + 7 x = 1 #. If one end of the line segment is at #(1 ,3 )#, where is the other end?

Answer 1

Any point on the line #-y+7x=-2#

Note that #(x,y)=(1,6)# is a point on the line -y+7x=1#
The distance from #(1,3)# to #(1,6)# is #3#
The distance from #(1,3)# to #(1,6+3)=(1,9)# is #6#
#(1,3), (1,6), and (1,9)# are co-linear.
Therefore the line segment from #(1,3)# to #(1,9)# is bisected by the line #-y+7x=1#
Furthermore a line segment between #(1,3)# and any point on a line through #(1,9)# parallel to #-y+7x=1# will also be bisected by #-y+7x=1#
The equation of this line is #color(white)("XXX")y-9=7(x-1)# or #color(white)("XXX")y-7x=2# or, in a form similar to the given equation #color(white)("XXX")-y+7x=-2#

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Answer 2

To find the other end of the line segment bisected by the line -y + 7x = 1, we first need to determine the slope of the line. The slope-intercept form of the equation of a line is y = mx + b, where m is the slope. Rearranging the given equation, we get y = 7x + 1.

The slope of the line is 7. Since the line bisects the line segment, the slope of the perpendicular bisector will be the negative reciprocal of 7, which is -1/7.

We also need the midpoint of the line segment, which can be found using the given endpoint and the fact that the line bisects it.

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

Given endpoint: (1, 3) Let the other endpoint be (x, y).

Using the midpoint formula, we have: [ \left( \frac{1 + x}{2}, \frac{3 + y}{2} \right) ]

The midpoint lies on the line -y + 7x = 1. So, we substitute the coordinates of the midpoint into the equation of the line:

[ -\frac{3 + y}{2} + 7 \left( \frac{1 + x}{2} \right) = 1 ]

[ -3 - y + 7 + 7x = 2 ]

[ 7x - y + 4 = 0 ]

Since this line passes through the midpoint of the line segment, the coordinates (x, y) satisfy the equation of the line.

We can solve the system of equations formed by the given endpoint and the equation of the line to find the coordinates of the other endpoint.

Substituting (1, 3) into the equation of the line, we get: [ -3 + 7(1) = 1 ] [ -3 + 7 = 1 ] [ 4 = 1 ]

This equation is false, which means (1, 3) does not lie on the line -y + 7x = 1. There might be an error in the question. Please double-check the information provided.

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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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