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

Question

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

Savannah Aguilar · Accepted Answer

The other end is at #(33/17,-106/17)#

Considering the following equation for the form's perpendicular bisector:

#Ax+By = C_1#

The bisected line's equation looks like this:

#Bx-Ay=C_2#

Using the above format to write the equation for the given perpendicular bisector:

#x+4y = 3#

We can now use the bisected line's general form:

#4x-y = C_2#

To find the value of #C_2#, we substitute the point #(5,6)#:

#4(5)-6 = C_2#

#C_2 = 14#

The bisected line's equation is as follows:

#4x-y = 14#

Determine where the lines intersect:

#x+4y = 3# #4x-y = 14#

Add to the first equation after multiplying the second equation by 4:

#17x= 59#

Calculate x:

#x = 59/17#

To solve for y, use the following line:

#59/17+4y = 3#

#y = -2/17#

The midpoint equations can be utilized to determine the opposite end:

#x_"mid" = (x_"start"+x_"end")/2#

#y_"mid" = (y_"start"+y_"end")/2#

Substitute #(5,6)# for the starts and #(59/17,-2/17)# for mids:

#59/17 = (5+x_"end")/2#

#-2/17 = (6+y_"end")/2#

Solve for #(x_"end", y_"end")#

#x_"end" = 33/17#

#y_"end" = -106/17#

The other end is at #(33/17,-106/17)#

Noah Atkinson · Answer

undefined To find the coordinates of the other end of the line segment, substitute $x = 5$ and $y = 6$ into the equation of the line, then solve for the other variable.

Given equation: $4y + x = 3$

Substitute $x = 5$ and $y = 6$:
$4(6) + 5 = 3$
$24 + 5 = 3$
$29 = 3$

This is a contradiction, which means there is no solution. There may be a mistake in the given information or in the calculation. Please double-check the equation or the provided coordinates.