A line segment is bisected by a line with the equation # 2 y + 3 x = 3 #. If one end of the line segment is at #( 1 , 8 )#, where is the other end?

Answer 1

To other end is at the point #(-83/13,40/13)#

The equation of any line segment that has the line,

#3x + 2y = 3" [1]"#

, as its perpendicular bisector will have the standard form:

#2x - 3y = C#

To find the value of C for the specified line segment, substitute the point #(1, 8)# into the equation:

#2(1) - 3(8) = C#

And then solve for C:

#C = -22#

The equation of the bisected line segment is:

#2x - 3y = -22" [2]"#

We shall use equations [1] and [2] to find the x coordinate of the point of intersection, #x_i#.

Multiply equation [1] by 3 and equation [2] by 2:

#9x_i + 6y_i = 9" [3]"#
#4x_i - 6y_i = -44" [4]"#

Add equation [3] to equation [4]:

#13x_i = -35#

#x_i = -35/13#

The change in x (#Deltax) #from the starting point to the point of intersection is as follows:

#Deltax = (-35/13 - 1)#

#Deltax = -48/13#

The x coordinate of the other end, #x_e# can be computed as follows:

#x_e = 2Deltax + x_s#

where #x_s# is the starting x coordinate, #x_s = 1#:

#x_e = 2(-48/13) + 1#

#x_e = -83/13#

To find the corresponding y coordinate, #y_e# substitute #x_e# into equation [2]:

#2(-83/13) - 3y_e = -22#

#y_e = 40/13#

To other end is at the point #(-83/13,40/13)#

Here is a graph with the two lines and the start and end points plotted:

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Answer 2

To find the coordinates of the other end of the line segment, we need to first find the coordinates of the point where the line segment is bisected. We can do this by solving the system of equations formed by the equation of the line and the coordinates of one end of the line segment.

Given: Equation of the line: 2y + 3x = 3 Coordinates of one end of the line segment: (1, 8)

Step 1: Substitute the x-coordinate of the given point into the equation of the line to find the corresponding y-coordinate. 3(1) + 2y = 3 3 + 2y = 3 2y = 3 - 3 2y = 0 y = 0

So, the coordinates of the point where the line segment is bisected are (1, 0).

Step 2: Calculate the difference between the y-coordinates of the given point and the point where the line segment is bisected. 8 - 0 = 8

Step 3: Add this difference to the y-coordinate of the point where the line segment is bisected to find the y-coordinate of the other end of the line segment. 0 + 8 = 8

So, the y-coordinate of the other end of the line segment is 8.

Step 4: Since the line segment is bisected, the x-coordinate of the other end is the same as the x-coordinate of the given point. So, the x-coordinate of the other end of the line segment is 1.

Therefore, the coordinates of the other end of the line segment are (1, 8).

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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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